Question 1: Palette 1/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): If an event $E$ can occur in $m$ ways and another independent event $F$ can occur in $n$ ways, then the total number of ways both events can occur together is $m \times n$.

There are 25 mathematics books and 24 physics books on a library shelf. In how many ways can we choose one mathematics and one physics book?

A

49

B

600

C

576

D

120

Question 2: Palette 2/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): If an event $E$ can occur in $m$ ways and an independent event $F$ can occur in $n$ ways, then the total number of ways both events can occur together is $m \times n$.

A dice is a six-faced cube, with the faces reading 1, 2, 3, 4, 5 and 6. When two dices are thrown we add the digits they show on top and take that sum as the result of the throw. In how many different ways the first throw of the 2 dices shows a total of 5, and second throw of the 2 dices shows a total of 4?

A

7

B

10

C

12

D

20

Question 3: Palette 3/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): If an event $E$ can occur in $m$ ways and an independent event $F$ can occur in $n$ ways, then the total number of ways both events can occur together is $m \times n$.

A firm is about to take on a new sales manager and a new receptionist. There are 6 applicants for the sales manager's job and 10 for that of the receptionist. How many ways can the pair be chosen?

A

16

B

50

C

60

D

600

Question 4: Palette 4/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): If a series of sequential choices need to be made, where the first choice can be made in $m_1$ ways, the second in $m_2$ ways, the third in $m_3$ ways, and so on, then the total number of ways to complete the sequence is $m_1 \times m_2 \times m_3 \times ... \times m_n$.

The Khanna family (Dad, Mum, Sia, and Jeet) move to a new district, and they decide that each person will register with a different dentist. There are 6 dentists in the district, how many ways can this family register?

A

120

B

240

C

360

D

720

Question 5: Palette 5/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): The number of ways to arrange $n$ distinct objects sequentially in $n$ positions is $n \times (n-1) \times (n-2) \times ... \times 1 = n!$.

Five children are running a race. How many different ways could they finish the race?

A

24

B

60

C

120

D

720

Question 6: Palette 6/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): If an event involves multiple independent choices, the total number of outcomes is the product of the number of choices available at each step.

How many integers are there less than 1000, ending with 3, 6 or 9?

A

270

B

300

C

333

D

1000

Question 7: Palette 7/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule): If a task consists of $k$ consecutive independent stages, with $n_1, n_2, ..., n_k$ choices at each stage, the total number of ways to perform the task is $n_1 \times n_2 \times ... \times n_k$.

A restaurant offers a choice of 3 salads, 5 main dishes, and 2 desserts. Use the fundamental principle of counting to find the number of different 3-course meals that can be selected.

A

10

B

25

C

30

D

60

Question 8: Palette 8/213

Comprehensive Algebra MCQ

Concept: Permutation of $n$ distinct items: The number of ways to arrange $n$ distinct objects in a line is given by $n! = n \times (n-1) \times ... \times 1$.

A teacher has 5 different books that he wishes to arrange in a row. How many different arrangements are possible?

A

25

B

60

C

120

D

243

Question 9: Palette 9/213

Comprehensive Algebra MCQ

Concept: Permutation of $r$ objects from $n$ distinct objects: The number of ordered arrangements of $r$ objects chosen from $n$ distinct objects is given by $P(n, r) = \frac{n!}{(n-r)!}$.

Suppose the teacher in Example 8 wishes to place only 3 of the 5 books in a row. How many arrangements of 3 books are possible?

A

15

B

60

C

120

D

243

Question 10: Palette 10/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Sum Rule and Multiplication Rule): If a task can be performed in mutually exclusive cases, the total number of ways is the sum of the number of ways for each case.

Suppose a hospital uses a light signal system with 4 distinct colours. A signal can be formed by lighting up 1, 2, 3, or 4 lights in a specific sequence. How many different signals can be produced?

A

24

B

32

C

64

D

60

Question 11: Palette 11/213

Comprehensive Algebra MCQ

Concept: Permutations and the Multiplication Principle of Counting. The total number of ways to arrange $r$ distinct items into $n$ distinct places is given by $P(n, r) = \frac{n!}{(n-r)!}$.

Four persons enter a railway carriage in which there are six seats. In how many ways can they take their places?

A

$120$

B

$360$

C

$720$

D

$240$

Question 12: Palette 12/213

Comprehensive Algebra MCQ

Concept: Permutation of $n$ distinct objects taken all at a time, which is given by $n! = n \times (n - 1) \times (n - 2) \times \dots \times 1$.

Four speakers expressed their desire to take the floor at a meeting. In how many ways is it possible to arrange them in the list of speakers?

A

$12$

B

$24$

C

$16$

D

$48$

Question 13: Palette 13/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule). If an event can occur in $m$ ways, followed by a second event in $n$ ways, and a third event in $p$ ways, then the total number of ways all three events can occur together is $m \times n \times p$.

There are four possible routes from police HQ to the bus station, four from the bus station to the bank, and four more from the bank to police HQ. How many ways can a police car patrol a circuit from police HQ to bus station to bank and back to HQ?

A

$12$

B

$48$

C

$64$

D

$256$

Question 14: Palette 14/213

Comprehensive Algebra MCQ

Concept: Permutation and the Multiplication Principle of Counting. When distinct items are placed without repetition, we use permutations $P(n, r) = \frac{n!}{(n-r)!}$. When repetition is allowed, each item has $n$ independent options, leading to $n^r$ total ways.

Four prisoners are being moved by train from one jail to another. If the train has eight coaches and each prisoner must travel in a different coach, in how many possible ways can the men travel on the train? How many ways would there be if they did not have to travel in different coaches?

A

$1680$ and $4096$

B

$4096$ and $1680$

C

$1680$ and $2401$

D

$6720$ and $4096$

Question 15: Palette 15/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule). If a task consists of several independent steps, the total number of ways to complete the task is the product of the number of ways to perform each individual step.

Certain registration numbers are formed from three different letters, chosen from the first ten letters of the alphabet, followed by a three digit number which must not begin with zero. Calculate how many registration numbers can be formed?

A

$720000$

B

$648000$

C

$583200$

D

$1000000$

Question 16: Palette 16/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting. When filling consecutive places, the choice for each place depends on excluding the digit used in the immediately preceding place while allowing digits used in earlier positions to be reused.

How many numbers of $n$ digits can be made with the non-zero digits in which no two consecutive digits are the same?

A

$9 \times 8^n$

B

$9^n$

C

$9 \times 8^{n-1}$

D

$8 \times 9^{n-1}$

Question 17: Palette 17/213

Comprehensive Algebra MCQ

Concept: Addition Principle of Counting (Sum Rule). If an event can occur in $m$ different ways, and another mutually exclusive event can occur in $n$ different ways, then the total number of ways in which either event can happen is $m + n$.

There are 30 persons in a group of which 20 are boys and 10 are girls. In how many ways can a group leader be selected if the group leader can either be a boy or a girl?

A

$200$

B

$30$

C

$20$

D

$10$

Question 18: Palette 18/213

Comprehensive Algebra MCQ

Concept: Addition Principle of Counting and Permutations. The total number of valid integers is found by summing the counts of 1-digit, 2-digit, and 3-digit integers with non-repeating digits.

How many of the first 1000 positive integers have distinct digits?

A

$738$

B

$648$

C

$720$

D

$810$

Question 19: Palette 19/213

Comprehensive Algebra MCQ

Concept: Addition Principle of Counting and Permutations/Multiplication Rule. The problem is solved by dividing into cases based on the number of digits (2-digit, 3-digit, and 4-digit numbers) and summing the results.

How many numbers between 10 and 10,000 can be formed by using the digits 1, 2, 3, 4, 5 if
(i) no digit is repeated in any number?
(ii) digits can be repeated?

A

(i) $200$, (ii) $775$

B

(i) $200$, (ii) $625$

C

(i) $180$, (ii) $775$

D

(i) $240$, (ii) $800$

Question 20: Palette 20/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting. The total number of distinct outfits formed by combining one item from each independent category is given by the product of the number of choices in each category.

A boy has five pairs of trousers and six shirts. His girlfriend has four skirts and seven tops. Who has the most choice of what to wear?

A

Boy ($30$ choices vs $28$ choices)

B

Girlfriend ($28$ choices vs $30$ choices)

C

Both have equal choices ($28$ choices each)

D

Both have equal choices ($30$ choices each)

Question 21: Palette 21/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting without repetition. Selecting the first action leaves one fewer choice for the second action.

A room has six doors. In how many ways is it possible to enter by one door and leave by another?

A

$36$

B

$30$

C

$25$

D

$11$

Question 22: Palette 22/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting (Multiplication Rule) for multiple independent selections.

How many different types of homes are available if a builder offers a choice of 5 basic plans, 3 roof styles, and 2 exterior finishes?

A

$10$

B

$25$

C

$30$

D

$60$

Question 23: Palette 23/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting across multiple independent attributes.

A tyre store carries 8 different sizes of tyres, each in both tube and tubeless variety, each with either nylon or rayon cord, and each with white sidewalls or plain black. How many different kinds of tyres does the store have?

A

$14$

B

$32$

C

$64$

D

$128$

Question 24: Palette 24/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting for combined product variations.

A mail order company offers 23 styles of ladies' slippers. If each style were available in 12 lengths, 3 widths and 6 colours, how many different kinds of ladies slippers would the warehouse have to keep in stock?

A

$4968$

B

$2484$

C

$1656$

D

$8280$

Question 25: Palette 25/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with positional constraints (thousands place cannot be 0, and units place determines evenness).

From the digits 0, 1, 2, 3, 4, 5, 6 how many four digit numbers can be constructed? How many of these are even numbers?

A

$2058$ and $1176$

B

$2401$ and $1372$

C

$2058$ and $1029$

D

$1715$ and $980$

Question 26: Palette 26/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle applied to fixed-length digit positions with restricted digit sets.

How many of the integers (whole numbers) between 10,000 and 100,000 have no digits other than 6, 7, or 8? How many have no digits other than 6, 7, 8 or 0?

A

$243$ and $768$

B

$243$ and $1024$

C

$125$ and $500$

D

$729$ and $768$

Question 27: Palette 27/213

Comprehensive Algebra MCQ

Concept: Counting ordered integer pairs under bounds and non-negativity restrictions.

How many solutions in positive integers x and y are there of the equation x + y = 100? By a solution, we mean an ordered pair (x, y) that satisfies the equation. How many solutions are there in non-negative integers?

A

$99$ and $101$

B

$100$ and $102$

C

$98$ and $100$

D

$99$ and $100$

Question 28: Palette 28/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting. The total number of terms in an expanded product of sums with distinct variables is equal to the product of the number of terms in each factor.

Find the number of terms in the product (a + b + c)(d + e + f)(p + q + r + s)(x + y + u + v + w).

A

$180$

B

$120$

C

$60$

D

$240$

Question 29: Palette 29/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting and Case Separation based on conditional choices.

A menu lists a choice of soup or orange juice for an appetizer, a choice of egg, chicken or fish for the dinner, and a choice of ice-cream or cake for dessert. A complete dinner consists of one choice in each case. Calculate: (i) How many different complete dinners are possible? (ii) How many complete dinners are there which have chicken for the dinner? (iii) How many complete dinners are available for a man who will eat cake only if he had egg?

A

(i) $12$, (ii) $4$, (iii) $8$

B

(i) $12$, (ii) $6$, (iii) $9$

C

(i) $18$, (ii) $6$, (iii) $8$

D

(i) $12$, (ii) $4$, (iii) $6$

Question 30: Palette 30/213

Comprehensive Algebra MCQ

Concept: Permutations of $n$ distinct objects taken $r$ at a time, given by the formula $P(n, r) = \frac{n!}{(n - r)!}$.

Suppose 8 people enter an event in a swim meet. In how many ways could the gold, silver, and bronze prizes be awarded?

A

$56$

B

$336$

C

$512$

D

$120$

Question 31: Palette 31/213

Comprehensive Algebra MCQ

Concept: Permutations of $n$ distinct objects taken $r$ at a time, given by the formula $P(n, r) = \frac{n!}{(n - r)!}$.

A group of students studies seven disciplines. In what ways is it possible to arrange the timetable for Monday if on this day of the week four lessons in different subjects are to take place?

A

$840$

B

$210$

C

$5040$

D

$35$

Question 32: Palette 32/213

Comprehensive Algebra MCQ

Concept: Permutations of distinct objects. The number of ways to select and arrange $r$ distinct items from a set of $n$ distinct items is given by $P(n, r) = \frac{n!}{(n - r)!}$.

How many integers between 100 and 999 inclusive consist of distinct odd digits?

A

$120$

B

$60$

C

$24$

D

$125$

Question 33: Palette 33/213

Comprehensive Algebra MCQ

Concept: Divisibility rules and Permutations. A number is a multiple of 5 if its last digit (units place) is 0 or 5. Arranging $n$ distinct objects taken $r$ at a time without repetition is given by $P(n, r) = \frac{n!}{(n - r)!}$.

How many six digit numbers multiple of 5 can be formed from the digits 1, 2, 3, 4, 5, 6, provided the numbers thus formed have no repeated digits?

A

$720$

B

$60$

C

$120$

D

$240$

Question 34: Palette 34/213

Comprehensive Algebra MCQ

Concept: Permutations without repetition and the Multiplication Principle of Counting. The total arrangements of $r$ distinct items from $n$ available items is given by $P(n, r) = \frac{n!}{(n-r)!}$. For conditional counting (such as even numbers), constraints are applied to specific digit positions first.

How many numbers of three digits can be formed using the digits 1, 2, 3, 4, 5, without repetition of digits? How many of these are even?

A

$60$ and $24$

B

$60$ and $36$

C

$120$ and $24$

D

$120$ and $48$

Question 35: Palette 35/213

Comprehensive Algebra MCQ

Concept: Permutations without repetition and the exclusion of numbers with a leading zero. A number between 100 and 1000 is a 3-digit number, where the hundreds place cannot be 0. The total number of valid permutations is $P(n, r) - P(n-1, r-1)$ where $0$ is fixed in the first position.

How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5, if repetition of the digits is not allowed?

A

$120$

B

$100$

C

$80$

D

$60$

Question 36: Palette 36/213

Comprehensive Algebra MCQ

Concept: Permutation formula $P(n, r) = \frac{n!}{(n - r)!}$ and algebraic simplification of factorials.

If $^{k+5}P_{k+1} = \frac{11(k - 1)}{2} \times {^{k+3}P_k}$, then find the value of $k$.

A

$k = 5$ or $k = 8$

B

$k = 6$ or $k = 7$

C

$k = 4$ or $k = 9$

D

$k = 3$ or $k = 6$

Question 37: Palette 37/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with conditional constraints. When forming a number with specific properties (such as being odd), fill the constrained position (units digit) first, followed by the remaining positions without repetition.

How many 4-digit numbers can be formed from 3, 4, 5, 6, 7 without repetition, if the numbers must all be odd?

A

$72$

B

$60$

C

$120$

D

$36$

Question 38: Palette 38/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with conditional constraints. When seating individuals with specific restrictions (such as driving ability), fill the constrained position (driver's seat) first, followed by the remaining positions.

Five friends are setting out on a journey in a 5-seater car. In how many ways can they seat themselves for the journey if only two of them can drive?

A

$48$

B

$120$

C

$24$

D

$60$

Question 39: Palette 39/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with positional restrictions and alternating patterns.

A pop-group's spot on a variety show is to consist of three vocal numbers and two instrumentals. In how many ways can the 'spot' be arranged so that it begins and ends with a vocal and neither instrumental follows directly after the other?

A

$12$

B

$24$

C

$36$

D

$18$

Question 40: Palette 40/213

Comprehensive Algebra MCQ

Concept: Addition Principle and Permutations without repetition ($P(n, r) = \frac{n!}{(n - r)!}$). The total count is found by considering valid numbers broken down by digit length (3-digit and 4-digit numbers).

Find the number of numbers between 300 and 3000 that can be formed with the digits 0, 1, 2, 3, 4 and 5, no digit being repeated in any number.

A

$180$

B

$60$

C

$120$

D

$240$

Question 41: Palette 41/213

Comprehensive Algebra MCQ

Concept: Permutations without repetition with constraints on specific positions. When counting even numbers where 0 is among the available digits, the calculation is split into cases based on whether 0 occupies the units place or another even digit occupies the units place (since 0 cannot occupy the thousands place).

How many even numbers of four digits can be formed with the digits 0, 1, 2, 3, 4, 5 and 6 without repetition?

A

$420$

B

$520$

C

$300$

D

$120$

Question 42: Palette 42/213

Comprehensive Algebra MCQ

Concept: Addition Principle and Permutations without repetition ($P(n, r) = \frac{n!}{(n - r)!}$). The problem is solved by splitting into cases based on the digit placed at the thousands place.

How many numbers of four digits greater than 2300 can be formed with the digits 0, 1, 2, 3, 4, 5 and 6; no digit being repeated in any number?

A

$560$

B

$480$

C

$520$

D

$600$

Question 43: Palette 43/213

Comprehensive Algebra MCQ

Concept: Addition Principle of Counting and Permutations without repetition ($P(n, r) = \frac{n!}{(n - r)!}$). The total count is found by considering valid numbers of different digit lengths (1-digit to 5-digit numbers) where the leading digit cannot be 0.

How many positive numbers can be formed by using any number of the digits 0, 1, 2, 3 and 4; no digit being repeated in any number?

A

$260$

B

$160$

C

$300$

D

$200$

Question 44: Palette 44/213

Comprehensive Algebra MCQ

Concept: Permutations without repetition ($P(n, r) = \frac{n!}{(n - r)!}$) and positional constraints. A number between 400 and 1000 must be a 3-digit number whose hundreds digit satisfies the given inequality.

How many numbers between 400 and 1000 can be made with the digits 2, 3, 4, 5, 6 and 0, repetition not allowed?

A

$120$

B

$60$

C

$90$

D

$45$

Question 45: Palette 45/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with non-zero restrictions on the leading digit. For a 3-digit number, the hundreds digit cannot be 0, while subsequent digits can include 0 as long as no digits repeat.

How many integers between 100 and 999 have distinct digits?

A

$729$

B

$648$

C

$504$

D

$656$

Question 46: Palette 46/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with conditional constraints. When counting numbers with specific constraints (such as being odd and having distinct digits), fill the constrained units place first, followed by the hundreds place (which cannot be 0), and then the tens place.

Of the 648 integers between 100 and 999 with distinct digits, how many are odd numbers?

A

$320$

B

$280$

C

$360$

D

$224$

Question 47: Palette 47/213

Comprehensive Algebra MCQ

Concept: Properties and expansion of factorials, where $n! = n \times (n-1) \times (n-2) \times \dots \times 1$. Factorial expressions can be simplified by factoring out common terms or expanding the larger factorial in terms of smaller factorials.

Simplify the following expressions:
(i) $\frac{7! - 5!}{4!4!}$
(ii) $\frac{(n + 1)!}{(n - 1)!}$
(iii) $\frac{(2n)!}{n!}$

A

(i) $\frac{103}{12}$, (ii) $n^2 + n$, (iii) $2^n \times [1 \times 3 \times 5 \times \dots \times (2n - 1)]$

B

(i) $\frac{205}{24}$, (ii) $n^2 - n$, (iii) $2^n \times n!$

C

(i) $\frac{103}{12}$, (ii) $n^2 + 1$, (iii) $2^n \times (2n - 1)!$

D

(i) $\frac{105}{24}$, (ii) $n^2 + n$, (iii) $2^{2n} \times n!$

Question 48: Palette 48/213

Comprehensive Algebra MCQ

Concept: Permutations of $n$ distinct objects taken all at a time is given by $n!$.

In an experiment on social interaction, 6 people will sit in 6 seats in a row. In how many ways can this be done?

A

$720$

B

$120$

C

$360$

D

$240$

Question 49: Palette 49/213

Comprehensive Algebra MCQ

Concept: Permutations of $n$ distinct items taken $r$ at a time without repetition, $P(n, r) = \frac{n!}{(n - r)!}$.

In how many ways can 3 men each be given a hotel room of his own if there are 8 free rooms available?

A

$56$

B

$512$

C

$336$

D

$120$

Question 50: Palette 50/213

Comprehensive Algebra MCQ

Concept: Arranging $r$ distinct items out of $n$ available distinct items over $r$ days without repetition, $P(n, r) = \frac{n!}{(n - r)!}$.

A man can cook 9 different dinners. How many different sequences of dinner could he cook for a week if he didn't prepare the same dinner twice?

A

$181440$

B

$362880$

C

$60480$

D

$15120$

Question 51: Palette 51/213

Comprehensive Algebra MCQ

Concept: Permutations of selecting and ordering $r$ distinct courses out of $n$ available courses, $P(n, r) = \frac{n!}{(n - r)!}$.

A business school offers courses in typing, shorthand, transcription, business english, technical writing, and accounting. In how many ways can a student arrange a schedule if 3 courses are taken?

A

$20$

B

$120$

C

$60$

D

$216$

Question 52: Palette 52/213

Comprehensive Algebra MCQ

Concept: Permutations of $n$ objects in $n$ fixed positions, $n!$.

In how many different ways can four passengers be seated in a railway coach compartment for four?

A

$16$

B

$12$

C

$24$

D

$48$

Question 53: Palette 53/213

Comprehensive Algebra MCQ

Concept: Permutations of selecting $r$ positions from $n$ distinct individuals where order matters, $P(n, r) = \frac{n!}{(n - r)!}$.

A trade union committee consists of 9 persons and has to elect a chairman, a vice-chairman and a treasurer. How many various combinations can there be?

A

$84$

B

$504$

C

$252$

D

$729$

Question 54: Palette 54/213

Comprehensive Algebra MCQ

Concept: Solving algebraic equations involving permutation formulas $P(n, r) = \frac{n!}{(n - r)!}$.

How many elements should be taken so that the number of their permutations taken 4 at a time is 6 times greater than those taken 2 elements at a time?

A

$6$

B

$4$

C

$5$

D

$7$

Question 55: Palette 55/213

Comprehensive Algebra MCQ

Concept: Permutations with fixed positional restrictions.

Determine the number of permutations of the letters in the word ROCKET. How many permutations can be made so that the letter R is the first letter and ET are the last two letters?

A

$720$ and $6$

B

$720$ and $24$

C

$120$ and $6$

D

$360$ and $12$

Question 56: Palette 56/213

Comprehensive Algebra MCQ

Concept: Permutations of choosing $r$ ordered roles from $n$ distinct members, $P(n, r) = \frac{n!}{(n - r)!}$.

In a club with 15 members, how many ways can a slate of 3 officers consisting of president, vice-president, and secretary/treasurer be chosen?

A

$455$

B

$2730$

C

$3375$

D

$1365$

Question 57: Palette 57/213

Comprehensive Algebra MCQ

Concept: Permutations of assigning players to distinct positions without repetition.

In how many ways can 5 players be assigned to the 5 positions on a basketball team, assuming that any player can play any position? In how many ways can 10 players be assigned to the 5 positions?

A

$120$ and $30240$

B

$120$ and $252$

C

$24$ and $30240$

D

$720$ and $15120$

Question 58: Palette 58/213

Comprehensive Algebra MCQ

Concept: Permutations of assigning $n$ distinct items into $k$ distinct slots ($k > n$) without repetition, represented by $P(k, n) = \frac{k!}{(k - n)!}$.

In how many ways can all of $n$ distinct objects be put in $k$ distinct boxes, not more than one in each box, if there are more boxes than things ($k > n$)?

A

$k^n$

B

$n^k$

C

$\frac{k!}{(k - n)!}$

D

$\frac{k!}{n!(k - n)!}$

Question 59: Palette 59/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition. Each distinct object (marble) can independently be placed into any of the available options (pockets). If there are $n$ options available for each of the $r$ objects, the total number of ways is $n^r$.

A child has four pockets and three marbles. In how many ways can the child put the marbles in its pockets?

A

$64$

B

$81$

C

$12$

D

$24$

Question 60: Palette 60/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition where each of the $r$ places can independently take any of the $n$ available choices ($n^r$).

A number lock has four rings, each with ten different digits (0 to 9). How many different attempts to open the lock might one need to make in order to find the correct permutation?

A

$1000$

B

$10000$

C

$5040$

D

$4000$

Question 61: Palette 61/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition for binary choices. For a sequence of length $r$ with $n$ possible symbols at each position, total arrangements = $n^r$.

The letters of the Morse code are sequences of dots and dashes. How many different letters can be formed by using 5 symbols?

A

$25$

B

$10$

C

$32$

D

$64$

Question 62: Palette 62/213

Comprehensive Algebra MCQ

Concept: Comparing permutations with repetition ($n^r$) and permutations without repetition ($P(n, r)$).

How many 3-letter code words (permutations) can be made from the word ANCHOR if:
(i) we can repeat letters?
(ii) we cannot repeat letters?

A

(i) $216$, (ii) $120$

B

(i) $120$, (ii) $216$

C

(i) $216$, (ii) $720$

D

(i) $120$, (ii) $60$

Question 63: Palette 63/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with restrictions on initial positions and repetition allowed.

How many car registration plates can be made, using two letters followed by four digits (e.g., RJ 6610)? (Zero is not allowed as the first of the four digits).

A

$6084000$

B

$6760000$

C

$5904900$

D

$608400$

Question 64: Palette 64/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition and counting non-empty subsets ($2^k - 1$).

A student appears in an objective test which contains 10 multiple choice questions. Each question has four choices and one correct answer.
(i) What is the maximum number of different answers can the student give?
(ii) How will the answer change if each question may have more than one correct answers?

A

(i) $4^{10}$, (ii) $15^{10}$

B

(i) $10^4$, (ii) $15^{10}$

C

(i) $4^{10}$, (ii) $16^{10}$

D

(i) $40$, (ii) $150$

Question 65: Palette 65/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition. Each toss of a coin has 2 possible outcomes (Heads or Tails). For $n$ independent tosses, the number of outcomes is $2^n$.

How many different outcomes are there in an experiment consisting of $n$ tosses of a coin?

A

$2^n$

B

$n^2$

C

$2n$

D

$n!$

Question 66: Palette 66/213

Comprehensive Algebra MCQ

Concept: Total number of possible combinations minus the single successful combination that opens the lock.

A letter lock consists of three rings each marked with fifteen different letters. Find in how many ways it is possible to make an unsuccessful attempt to open the lock?

A

$3374$

B

$3375$

C

$4095$

D

$2743$

Question 67: Palette 67/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition where each of the $r$ stalls can independently take any of the $n$ available types of animals ($n^r$).

There are stalls for 12 animals in a ship. In how many ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 12?

A

$3^{12}$

B

$12^3$

C

$P(12, 3)$

D

$3! \times 12!$

Question 68: Palette 68/213

Comprehensive Algebra MCQ

Concept: Permutation with repetition where each object (delegate) can independently choose any option (hotel), giving $n^r$ total ways.

In how many ways 5 delegates can be put in 6 hotels of a city if there is no restriction?

A

$6^5$

B

$5^6$

C

$P(6, 5)$

D

$6! \times 5!$

Question 69: Palette 69/213

Comprehensive Algebra MCQ

Concept: Permutations with repetition and conditional choice reduction for consecutive items.

In how many ways can a ten question multiple choice examination be answered if there are four choices a, b, c and d to each question? If no two consecutive questions are answered the same way, how many ways are there?

A

$4^{10}$ and $4 \times 3^9$

B

$4^{10}$ and $3^{10}$

C

$10^4$ and $10 \times 3^9$

D

$4^{10}$ and $4! \times 3^9$

Question 70: Palette 70/213

Comprehensive Algebra MCQ

Concept: Calculating the total number of permutations with repetition ($10^5$) and converting time units to compare against available working hours.

A safe is locked by a device consisting of five disks with the digits 0, 1, 2, ..., 9 on each of them. The safe gets unlocked by dialing a certain combination of digits. Will ten days be enough to open the safe if the working day lasts 13 hours and it takes five seconds to dial one combination of digits?

A

No, because it requires about 138.89 working hours, which exceeds 130 hours.

B

Yes, because it takes only 100 working hours.

C

Yes, because it takes 120 working hours.

D

No, because it requires 200 working hours.

Question 71: Palette 71/213

Comprehensive Algebra MCQ

Concept: Sum of permutations with repetition for 1-digit through 7-digit numbers using 10 possible digits (0-9).

How many different telephone numbers are there if it is assumed that each number contains not more than seven digits (a telephone number may begin with a zero)?

A

$11111110$

B

$10000000$

C

$9999999$

D

$11111100$

Question 72: Palette 72/213

Comprehensive Algebra MCQ

Concept: Comparing unrestricted arrangement ($n^r$), arrangement without repetition ($P(n, r)$), and complementary counting ($n^r - n$).

In how many ways can 5 delegates be put in 6 hotels of a city if:
(i) there is no restriction?
(ii) no two delegates can stay together?
(iii) all the delegates do not stay in the same hotel?

A

(i) $7776$, (ii) $720$, (iii) $7770$

B

(i) $7776$, (ii) $120$, (iii) $7770$

C

(i) $3125$, (ii) $720$, (iii) $3119$

D

(i) $7776$, (ii) $720$, (iii) $7716$

Question 73: Palette 73/213

Comprehensive Algebra MCQ

Concept: Counting options per issue with repetition allowed, minus the fully restricted case (abstaining on all issues).

The members of a club are to vote "yes" or "no" on each of eight issues. In marking his ballot, a member has the option of abstaining on as many as seven of the issues, but he should not abstain in all eight cases. In how many ways can a ballot be marked?

A

$6560$

B

$6561$

C

$255$

D

$256$

Question 74: Palette 74/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting. Selecting winning students from $20$ available candidates for distinct prizes across different subjects.

In a school there are two prizes for excellence in Mathematics (1st and 2nd), two in Chemistry (1st and 2nd), and only one in Physics (1st). In how many ways can these prizes be awarded to 20 students?

A

$3040000$

B

$3200000$

C

$1520000$

D

$3800000$

Question 75: Palette 75/213

Comprehensive Algebra MCQ

Concept: Permutations with specific positional constraints, both with and without repetition.

How many different 4-letter radio station call letters can be made:
(a) if the first letter must be K or W and no letter may be repeated?
(b) if repeats are allowed (but the first letter is K or W)?
(c) How many of the 4-letter call letters (starting with K or W) with no repeats end in R?

A

(a) $27600$, (b) $35152$, (c) $1104$

B

(a) $27600$, (b) $35152$, (c) $2208$

C

(a) $13800$, (b) $17576$, (c) $1104$

D

(a) $24000$, (b) $35152$, (c) $1200$

Question 76: Palette 76/213

Comprehensive Algebra MCQ

Concept: Multiplication principle combining letters ($26$) and digits ($10$), with and without repetition.

For many years, Delhi used 3 letters followed by 3 digits on its automobile license plates.
(a) How many different license plates are possible with this arrangement?
(b) How many plates are possible if no letter or digit can be repeated on a single plate?

A

(a) $17576000$, (b) $11232000$

B

(a) $17576000$, (b) $12000000$

C

(a) $15600000$, (b) $11232000$

D

(a) $17576000$, (b) $15600000$

Question 77: Palette 77/213

Comprehensive Algebra MCQ

Concept: Permutations with repetition ($n^r$). Each ring can independently be worn on any of the available fingers, so for $r$ rings and $n$ fingers, the total number of ways is $n^r$.

In how many ways can 6 rings be worn on the four fingers of one hand?

A

$4096$

B

$1296$

C

$24$

D

$360$

Question 78: Palette 78/213

Comprehensive Algebra MCQ

Concept: Combining unrestricted placement ($n^r$), placement without repetition ($P(n, r)$), and complementary counting ($n^r - \text{restricted cases}$).

In how many ways four friends can put up in 8 hotels of a town if
(i) There is no restriction?
(ii) No two friends can stay together?
(iii) All the friends do not stay in same hotel?

A

(i) $4096$, (ii) $1680$, (iii) $4088$

B

(i) $4096$, (ii) $1680$, (iii) $4092$

C

(i) $6561$, (ii) $1680$, (iii) $6553$

D

(i) $4096$, (ii) $336$, (iii) $4088$

Question 79: Palette 79/213

Comprehensive Algebra MCQ

Concept: Sum of digits in a specific place value across all permutations. If $n$ distinct non-zero digits are taken all at a time without repetition, each digit appears in the unit place $(n - 1)!$ times. The sum of digits at the unit place is given by $(n - 1)! \sum d_i$.

Find the sum of the digits in the unit place of all numbers formed with the help of 3, 4, 5, 6 taken all at a time.

A

$108$

B

$72$

C

$144$

D

$96$

Question 80: Palette 80/213

Comprehensive Algebra MCQ

Concept: Sum of numbers formed using $n$ distinct digits including 0. The total sum is found by calculating the contribution of non-zero digits across all place values, accounting for the condition that 0 cannot occupy the thousands place.

Find the sum of all the four-digit numbers that can be formed with the digits 0, 1, 2 and 3 (repetition of digits not allowed).

A

$39996$

B

$38664$

C

$36000$

D

$32448$

Question 81: Palette 81/213

Comprehensive Algebra MCQ

Concept: Sum of numbers formed using digits with repetitions. Calculating the frequency of each distinct digit at each place value using permutations of multiset items.

Find the sum of all the 4-digit numbers that can be formed with the digits 1, 2, 2 and 3.

A

$18884$

B

$24444$

C

$28884$

D

$26664$

Question 82: Palette 82/213

Comprehensive Algebra MCQ

Concept: Sum of all $n$-digit numbers formed using $n$ distinct non-zero digits without repetition:
$\text{Sum} = (n - 1)! \times (\sum d_i) \times \left(\frac{10^n - 1}{9}\right)$.

Find the sum of all five-digit numbers that can be formed using the digits 1, 2, 3, 4 and 5 (repetition of digits not allowed).

A

$4111110$

B

$3199960$

C

$3999960$

D

$3888860$

Question 83: Palette 83/213

Comprehensive Algebra MCQ

Concept: Recurrence relation for permutations. Counting $r$-permutations of $n$ distinct items by partitioning into two disjoint cases: permutations that contain a specified item versus permutations that do not contain that specified item.

Prove that $P(n, r) = r \times P(n - 1, r - 1) + P(n - 1, r)$. Which of the following correctly describes this algebraic identity?

A

Partitioning combinations of $n$ items into two equal subsets.

B

Partitioning permutations of $n$ items into those containing a specific item ($r \times P(n - 1, r - 1)$) and those not containing it ($P(n - 1, r)$).

C

Applying Pascal's identity to factorial expansions.

D

Permutations with repetition allowed across $r$ positions.

Question 84: Palette 84/213

Comprehensive Algebra MCQ

Concept: Sum of permutations $P(n, k)$ for varying lengths $k = 1, 2, \dots, n$.

Find the total number of signals that can be made by five flags of different colours when any number of them may be used.

A

$240$

B

$120$

C

$325$

D

$720$

Question 85: Palette 85/213

Comprehensive Algebra MCQ

Concept: Counting arrangements with restrictions using complementary counting and inclusion-exclusion principle for distinct colour requirements.

There are counters available in 3 different colours (at least four of each colour). Counters are all alike except for the colour. If $m$ denotes the number of arrangements of four counters if no arrangement consists of counters of same colour and $n$ denotes the corresponding figure when every arrangement consists of counters of each colour, then find the value of $m / n$.

A

$\frac{78}{35}$

B

$\frac{26}{9}$

C

$\frac{12}{5}$

D

$\frac{13}{6}$

Question 86: Palette 86/213

Comprehensive Algebra MCQ

Concept: Permutations and Lexicographical Ranking

If all the letters of the word QUEST are arranged in all possible ways and put in dictionary order, then find the rank of the given word.

A

41

B

42

C

43

D

44

Question 87: Palette 87/213

Comprehensive Algebra MCQ

Concept: Permutations and Lexicographical Ranking

The letters of the word OUGHT are written in all possible orders and these words rewritten out as in a dictionary. Find the rank of the word TOUGH in this dictionary.

A

89

B

87

C

91

D

85

Question 88: Palette 88/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting and Parity Analysis

Find the number of 7-digit numbers the sum of whose digits is even.

A

$45 \times 10^4$

B

$9 \times 10^5$

C

$45 \times 10^5$

D

$10^6$

Question 89: Palette 89/213

Comprehensive Algebra MCQ

Concept: Multiplication Principle of Counting with Restrictions

A new club flag is to be designed with 6 vertical stripes using some or all of the colours yellow, green, blue and red. In how many ways can this be done so that no two adjacent stripes have the same colour?

A

$4 \times 3^5$

B

$4^6$

C

$3^6$

D

$4 \times 3^6$

Question 90: Palette 90/213

Comprehensive Algebra MCQ

Concept: Permutations with Repetition

How many different five-digit numbers are there (leading zeros not allowed)?

A

$9 \times 10^4$

B

$10^5$

C

$9 \times 10^5$

D

$9 \times 9^4$

Question 91: Palette 91/213

Comprehensive Algebra MCQ

Concept: Counting Principles for Even Numbers

How many even 5-digit numbers are there?

A

$45 \times 10^4$

B

$45 \times 10^3$

C

$9 \times 10^4$

D

$5 \times 10^4$

Question 92: Palette 92/213

Comprehensive Algebra MCQ

Concept: Palindromic Numbers and Counting Principle

How many 5-digit numbers are there that are the same when the order of their digits is reversed (e.g., 14341)?

A

$9 \times 10^3$

B

$9 \times 10^2$

C

$10^3$

D

$9 \times 10^4$

Question 93: Palette 93/213

Comprehensive Algebra MCQ

Concept: Permutations and Counting Principles

Find the number of positive integers with distinct digits.

A

$8877690$

B

$9876540$

C

$8123450$

D

$7654320$

Question 94: Palette 94/213

Comprehensive Algebra MCQ

Concept: Divisibility Rule of 9 and Permutations

Find the number of 9 digit numbers divisible by 9 which can be formed using the digits from 0 to 9 without repetition of digits.

A

$322560$

B

$362880$

C

$685440$

D

$725760$

Question 95: Palette 95/213

Comprehensive Algebra MCQ

Concept: Sum of Numbers Formed by Permutation of Digits

Find the sum of all the four digit numbers that can be formed with the digits 3, 2, 3, 4.

A

$39996$

B

$36669$

C

$38886$

D

$41112$

Question 96: Palette 96/213

Comprehensive Algebra MCQ

Concept: Sum of Digits at Place Values

Find the sum of all numbers greater than 10,000 formed with the digits 0, 2, 4, 6 and 8; no digit being repeated in any number.

A

$4800000$

B

$5199960$

C

$5332800$

D

$4999960$

Question 97: Palette 97/213

Comprehensive Algebra MCQ

Concept: Sum of Permutations of Digits with Zero Present

Find the sum of all the 4 digit numbers that can be formed with the digits 0, 2, 3 and 5.

A

$64440$

B

$58880$

C

$66660$

D

$62220$

Question 98: Palette 98/213

Comprehensive Algebra MCQ

Concept: Lexicographical Order and Rank of a Word

The letters of the word RANDOM are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word RANDOM.

A

$600$

B

$614$

C

$620$

D

$608$

Question 99: Palette 99/213

Comprehensive Algebra MCQ

Concept: Linear Permutation and Relative Position Constraints

In how many ways can six persons A, B, C, D, E, F line up at a railway booking window? In how many of these A stands first? In how many of these A gets the ticket before B?

A

$720, 120, 360$

B

$720, 240, 360$

C

$5040, 720, 2520$

D

$360, 60, 180$

Question 100: Palette 100/213

Comprehensive Algebra MCQ

Concept: Restricted Permutations and Combinations with/without Repetition

How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, if: (i) repetition of digits is allowed? (ii) repetition of digits is not allowed?

A

$375, 72$

B

$376, 72$

C

$376, 96$

D

$375, 96$

Question 101: Palette 101/213

Comprehensive Algebra MCQ

Concept: Divisibility Rule of 3 and Combinatorial Selection

How many five digits numbers divisible by 3 can be formed using the digits 0, 1, 2, 3, 4, 7 and 8 if each digit is to be used at most once.

A

$720$

B

$744$

C

$680$

D

$756$

Question 102: Palette 102/213

Comprehensive Algebra MCQ

Concept: Fundamental Multiplication Principle of Counting

In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?

A

$86400$

B

$72000$

C

$43200$

D

$51840$

Question 103: Palette 103/213

Comprehensive Algebra MCQ

Concept: Forming Numbers with Specific Magnitude Constraints

How many different numbers greater than 5000 can be formed with the digits 0, 1, 5, 9, no digit being repeated?

A

$24$

B

$12$

C

$18$

D

$16$

Question 104: Palette 104/213

Comprehensive Algebra MCQ

Concept: Counting Numbers in Given Range

Find the number of numbers lying between 300 and 4000 that can be formed with the digits 0, 1, 2, 3, 4, 5; no digit being repeated.

A

$240$

B

$180$

C

$210$

D

$300$

Question 105: Palette 105/213

Comprehensive Algebra MCQ

Concept: Parity of Digit Sums and Fundamental Counting Rule

How many five digits numbers are there whose sum of digits is odd?

A

$45000$

B

$50000$

C

$40000$

D

$90000$

Question 106: Palette 106/213

Comprehensive Algebra MCQ

Concept: Divisibility Rules and Position Constraints

If repetition of digits is not allowed how many numbers of four digits divisible by 5 can be formed with the digits 0, 4, 5, 6, 7?

A

$42$

B

$48$

C

$36$

D

$24$

Question 107: Palette 107/213

Comprehensive Algebra MCQ

Concept: Divisibility by 25 without Repetition

Find the number of numbers divisible by 25 that can be formed using only the digits 1, 2, 3, 4, 5 and 0 taken five at a time.

A

$42$

B

$36$

C

$48$

D

$54$

Question 108: Palette 108/213

Comprehensive Algebra MCQ

Concept: Combinations vs Permutations (Selection vs Ordering)

A sales representative has 10 accounts in a certain city.
(a) In how many ways can 3 accounts be selected to call on?
(b) In how many ways can calls be scheduled for 3 of the 10 accounts?

A

$120$ and $720$

B

$720$ and $120$

C

$210$ and $504$

D

$120$ and $120$

Question 109: Palette 109/213

Comprehensive Algebra MCQ

Concept: Combination of 2 Elements from $n$ Distinct Elements

How many matches will be played in a football championship of 16 participants if every two teams play with each other once?

A

$256$

B

$120$

C

$240$

D

$160$

Question 110: Palette 110/213

Comprehensive Algebra MCQ

Concept: Selection of $r$ Objects from $n$ Objects

Two ambulance attendants are needed in any emergency and there are six men on duty. In how many ways can the two be chosen?

A

$30$

B

$12$

C

$15$

D

$36$

Question 111: Palette 111/213

Comprehensive Algebra MCQ

Concept: Division into Groups / Selecting Subsets

In how many ways can 7 different books be given to two students if one student is to have 4 books and the other is to have 3 books?

A

$42$

B

$35$

C

$70$

D

$210$

Question 112: Palette 112/213

Comprehensive Algebra MCQ

Concept: Combinatorial Selection and Inclusion Constraints

A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. Find the number of children in her class.

A

$10$

B

$12$

C

$8$

D

$15$

Question 113: Palette 113/213

Comprehensive Algebra MCQ

Concept: Properties of Combinations and Factorials

Prove that the product of $r$ consecutive integers is divisible by $r!$.

A

$p = \frac{r!}{n!}$

B

$\frac{p}{r!} = ^{n+r-1}C_r$ which is an integer

C

$\frac{p}{r!} = ^nC_r$

D

$p = r! \times n!$

Question 114: Palette 114/213

Comprehensive Algebra MCQ

Concept: Algebraic Equations Involving Permutations and Combinations

Solve the equation $^{x+3}P_2 = ^{x+2}C_3 + 20$.

A

$x = 2$

B

$x = 5$

C

$x = 3$

D

$x = 4$

Question 115: Palette 115/213

Comprehensive Algebra MCQ

Concept: Division of Distinct Objects into Specified Group Sizes

Suppose we have 10 social workers and we wish them to work in groups of 2, 3 and 5. How many ways can we split them up into groups of the right size?

A

$2520$

B

$1260$

C

$5040$

D

$3150$

Question 116: Palette 116/213

Comprehensive Algebra MCQ

Concept: Partitioning a Set into Distinct Named Groups

In this year's school play the cast consists of 5 members of a royal family, 15 soldiers, and 30 peasants. How many ways can the cast of 50 pupils be divided among these three groups?

A

$^{50}C_5 \times ^{45}C_{15}$

B

$^{50}C_{30} \times ^{20}C_{15}$

C

$\frac{50!}{5! 15! 30!}$

D

All of the above

Question 117: Palette 117/213

Comprehensive Algebra MCQ

Concept: Combinations with Repetition and Identical Objects on Restricted Positions

How many ways can 12 identical white pawns and 12 identical black pawns be placed on the black squares of an $8 \times 8$ chessboard?

A

$^{32}C_{12} \times ^{20}C_{12}$

B

$^{64}C_{12} \times ^{52}C_{12}$

C

$^{32}C_{24}$

D

$^{32}C_{12} \times ^{12}C_{12}$

Question 118: Palette 118/213

Comprehensive Algebra MCQ

Concept: Counting Pairs in Rows and Columns

How many ways are there to place 2 identical rooks in a common row or column of an $8 \times 8$ chessboard?

A

$16 \times ^8C_2$

B

$8 \times ^8C_2$

C

$2 \times 8 \times ^8C_2$

D

$64 \times ^8C_2$

Question 119: Palette 119/213

Comprehensive Algebra MCQ

Concept: Complementary Counting on Restricted Grids

How many ways are there to place 2 identical kings on an $8 \times 8$ chessboard so that the kings are not in adjacent squares in the same row or column?

A

$2 \times 8 \times (^8C_2 - 7)$

B

$16 \times ^8C_2 - 56$

C

$2 \times 8 \times (^8C_2 - 8)$

D

$8 \times (^8C_2 - 7)$

Question 120: Palette 120/213

Comprehensive Algebra MCQ

Concept: Combinations and Modular Arithmetic / Complementary Counting

All possible two-factor products are formed from the numbers 1, 2, 3, ..., 100. How many numbers out of the total obtained are multiples of 3?

A

$2211$

B

$2739$

C

$4950$

D

$2310$

Question 121: Palette 121/213

Comprehensive Algebra MCQ

Concept: Fundamental Principle of Counting and Combinations

A basketball coach must select two attackers and two defenders from among three attackers and five defenders. How many different combinations of attackers and defenders can he select?

A

$30$

B

$15$

C

$20$

D

$25$

Question 122: Palette 122/213

Comprehensive Algebra MCQ

Concept: Addition and Multiplication Principles in Combinatorial Selection

A cricket team of 11 men is to be chosen from 15 men. Four of the 15 can bowl and two others can keep wicket. If the team is must include only one wicket keeper and at least three bowlers, in how many ways can it be chosen?

A

$456$

B

$512$

C

$384$

D

$420$

Question 123: Palette 123/213

Comprehensive Algebra MCQ

Concept: Case Analysis and Combination Sum Rule

A group of students consists of 4 boys and 5 girls. Find the number of ways of selecting a team of at least 3 boys and 4 girls.

A

$25$

B

$30$

C

$35$

D

$40$

Question 124: Palette 124/213

Comprehensive Algebra MCQ

Concept: Pascal's Identity for Combinations

Evaluate $^nC_r + \sum_{j=0}^{3} {^{n+j}C_{r+1+j}}$.

A

$^{n+3}C_{r+3}$

B

$^{n+4}C_{r+4}$

C

$^{n+4}C_{r+3}$

D

$^{n+3}C_{r+4}$

Question 125: Palette 125/213

Comprehensive Algebra MCQ

Concept: Maximum Value of Combination $^{n}C_r$ for Odd $n$

A class has 21 students. The class teacher has been asked to make groups of $r$ students each and go to the zoo taking one group at a time. Find the size of the group for which the teacher goes the maximum number of times to the zoo.

A

10 or 11

B

9 or 12

C

11 or 12

D

8 or 10

Question 126: Palette 126/213

Comprehensive Algebra MCQ

Concept: Ratio Properties of Combinations and Factorial Simplification

If $^{n+1}C_{r+1} : ^nC_r : ^{n-1}C_{r-1} = 11 : 6 : 3$, then find the value of $nr$.

A

$30$

B

$50$

C

$40$

D

$60$

Question 127: Palette 127/213

Comprehensive Algebra MCQ

Concept: Consecutive Combination Ratios and System of Linear Equations

If $^{n}C_{r-1} = 36$, $^nC_r = 84$, and $^{n}C_{r+1} = 126$, then find $r$.

A

$4$

B

$5$

C

$3$

D

$2$

Question 128: Palette 128/213

Comprehensive Algebra MCQ

Concept: Vandermonde's Identity and Combinatorial Proof

Prove or identify the correct algebraic form of Vandermonde's Identity: $^{n+m}C_r = \sum_{k=0}^{r} {^nC_k \cdot ^mC_{r-k}}$.

A

$^nC_0 \cdot ^mC_r + ^nC_1 \cdot ^mC_{r-1} + \dots + ^nC_r \cdot ^mC_0$

B

$^nC_r + ^mC_r$

C

$^nC_r \cdot ^mC_r$

D

$\sum_{k=0}^{r} {^nC_k \cdot ^mC_k}$

Question 129: Palette 129/213

Comprehensive Algebra MCQ

Concept: Combinations and Elementary Probability Principles with Playing Cards

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these:
(i) four cards are of the same suit,
(ii) four cards belong to four different suits,
(iii) are face cards,
(iv) two are red cards and two are black cards,
(v) cards are of the same color?

A

Total: $270725$; (i) $2860$, (ii) $13^4$, (iii) $495$, (iv) $105625$, (v) $29900$

B

Total: $270725$; (i) $2860$, (ii) $13^4$, (iii) $495$, (iv) $29900$, (v) $105625$

C

Total: $270725$; (i) $1430$, (ii) $13^4$, (iii) $495$, (iv) $105625$, (v) $29900$

D

Total: $270725$; (i) $2860$, (ii) $52^4$, (iii) $495$, (iv) $105625$, (v) $29900$

Question 130: Palette 130/213

Comprehensive Algebra MCQ

Concept: Permutation (Order Matters) vs Combination (Order Does Not Matter)

Which of the following everyday activities involve permutations, and which involve combinations only?
(i) Choosing a cricket team from a class of 24 boys.
(ii) Deciding which of the boys will bat first, second, and so on.
(iii) Deciding which three books to borrow from a library.
(iv) Planning the best sequence in which to decorate the rooms in my house.
(v) Planting a row of rose-bushes so that red and yellow flowers alternate.

A

(i) Combination, (ii) Permutation, (iii) Combination, (iv) Permutation, (v) Permutation

B

(i) Permutation, (ii) Combination, (iii) Combination, (iv) Permutation, (v) Permutation

C

(i) Combination, (ii) Permutation, (iii) Permutation, (iv) Combination, (v) Combination

D

(i) Permutation, (ii) Permutation, (iii) Combination, (iv) Combination, (v) Permutation

Question 131: Palette 131/213

Comprehensive Algebra MCQ

Concept: Geometric Combinations from Vertices

How many triangles can be constructed by connecting the vertices of an octagon?

A

$48$

B

$56$

C

$64$

D

$336$

Question 132: Palette 132/213

Comprehensive Algebra MCQ

Concept: Selection of Objects (Combinations)

How many ways can a traffic warden with 3 parking tickets left in his book, give them to three out of the seven cars he find that have over-parked?

A

$210$

B

$42$

C

$35$

D

$21$

Question 133: Palette 133/213

Comprehensive Algebra MCQ

Concept: Forming Subsets of Equal Size

A party of eight people arrive at a river and a boatman agrees to take them across in two groups of four. In how many ways can the first boatload be chosen?

A

$70$

B

$1680$

C

$35$

D

$140$

Question 134: Palette 134/213

Comprehensive Algebra MCQ

Concept: Combinatoric Selection and Time Intervals

In a local police station, a different combination of three constables is selected daily for duty. If there are ten constables available and the system operates every day except Sunday, how many weeks will go by before the same three constables are on duty again together?

A

$10$ weeks

B

$20$ weeks

C

$12$ weeks

D

$15$ weeks

Question 135: Palette 135/213

Comprehensive Algebra MCQ

Concept: Combinations and Modular Arithmetic / Complementary Counting

All possible two factor products are formed from the numbers 1, 2, 3, ..., 100. How many numbers out of the total obtained are multiples of 3?

A

$2211$

B

$2739$

C

$4950$

D

$2310$

Question 136: Palette 136/213

Comprehensive Algebra MCQ

Concept: Domain Constraints of Combinations $^nC_r$

Find the domain of definition of the function $f(x) = ^{2x-8}C_{x+1}$ and the set of its values.

A

Domain: $\{9, 10\}$, Set of values: $\{10, 66\}$

B

Domain: $\{9, 10, 11\}$, Set of values: $\{10, 45, 66\}$

C

Domain: $\{8, 9, 10\}$, Set of values: $\{8, 10, 66\}$

D

Domain: $\{9, 10\}$, Set of values: $\{10, 45\}$

Question 137: Palette 137/213

Comprehensive Algebra MCQ

Concept: Ratio of Consecutive Combinations

Solve the given inequalities:
(a) $^{10}C_{x-1} > 2 \cdot ^{10}C_x$
(b) $8 \cdot ^{105}C_x < 3 \cdot ^{105}C_{x+1}$

A

(a) $x \ge 8$, (b) $x < 28$

B

(a) $x \ge 8$, (b) $x > 28$

C

(a) $x \le 7$, (b) $x < 28$

D

(a) $x \ge 8$, (b) $x > 27$

Question 138: Palette 138/213

Comprehensive Algebra MCQ

Concept: Combinatorial Inequality Solving

Find the positive integral values of $x$ such that $^{x-1}C_4 - ^{x-1}C_3 - \frac{5}{4}(x-2)(x-3) < 0$.

A

$x \in \{5, 6, 7, 8, 9\}$

B

$x \in \{5, 6, 7, 8\}$

C

$x \in \{1, 2, 3, 4, 5\}$

D

$x \in \{5, 6, 7\}$

Question 139: Palette 139/213

Comprehensive Algebra MCQ

Concept: Multinomial Partitioning into Unassigned Groups

Twelve people are waiting for a bus. When it comes it only has room for 6 on top and 4 downstairs. So in how many ways could you form the three groups (those who go upstairs, downstairs or wait)?

A

$138600$

B

$277200$

C

$11550$

D

$13860$

Question 140: Palette 140/213

Comprehensive Algebra MCQ

Concept: Combinatorial Equations in Game Tournaments

There are $m$ men and 2 women participating in a chess tournament. Every participant plays two games with every other participant. If the number of games played by the men between themselves exceeds by 66 the number of games played between men and women, then find the value of $m$ and the total number of games played in the tournament.

A

$m = 11$, total games = $156$

B

$m = 12$, total games = $182$

C

$m = 10$, total games = $132$

D

$m = 9$, total games = $110$

Question 141: Palette 141/213

Comprehensive Algebra MCQ

Concept: Selection of Objects from Distinct Subsets

A bag contains 5 black and 6 red balls, all balls being different. Determine the number of ways in which 2 black and 3 red balls can be selected.

A

$200$

B

$150$

C

$100$

D

$250$

Question 142: Palette 142/213

Comprehensive Algebra MCQ

Concept: Sequential Selection with Permutation and Combination

A meeting of 40 people must choose a chairman, a secretary and 5 members of a committee. How many different committees can be formed?

A

$^{40}P_2 \times ^{38}C_5$

B

$^{40}C_7 \times 7$

C

$^{40}C_2 \times ^{38}C_5$

D

$^{40}P_7$

Question 143: Palette 143/213

Comprehensive Algebra MCQ

Concept: Independent Combinatorial Choices

A team consists of two house painters, three plasterers and one joiner. How many different teams can be formed from a staff of fifteen house painters, ten plasterers and five joiners?

A

$63000$

B

$31500$

C

$126000$

D

$15750$

Question 144: Palette 144/213

Comprehensive Algebra MCQ

Concept: Complementary Counting for "At Least One"

Ten cards have been taken from a pack of 52. In how many cases will there be at least one ace among the selected cards?

A

$^{52}C_{10} - ^{48}C_{10}$

B

$^{48}C_{10}$

C

$^{52}C_{10} - ^4C_1 \times ^{48}C_9$

D

$^4C_1 \times ^{48}C_9$

Question 145: Palette 145/213

Comprehensive Algebra MCQ

Concept: Selection with Dependent Suit Restrictions

How many ways are there of choosing a hand of 6 cards containing an ace and a king of the same suit from a pack of 52 cards?

A

$4 \times ^{50}C_4$

B

$4 \times ^{48}C_4$

C

$^{52}C_6 - ^{48}C_6$

D

$13 \times ^{48}C_4$

Question 146: Palette 146/213

Comprehensive Algebra MCQ

Concept: Constrained Category Selection

In how many ways can a cricket team be selected from a group of 25 players containing 10 batsmen, 8 bowlers, 5 all-rounders and 2 wicket keepers? Assume that the team of 11 players requires 5 batsmen, 3 all-rounders, 2 bowlers and 1 wicket keeper.

A

$141120$

B

$70560$

C

$282240$

D

$142000$

Question 147: Palette 147/213

Comprehensive Algebra MCQ

Concept: Permutations of Objects Not All Distinct (Alike Objects)

In how many ways can the letters of the word DADDY be permuted among themselves?

A

60

B

120

C

20

D

24

Question 148: Palette 148/213

Comprehensive Algebra MCQ

Concept: Permutations of Objects Not All Distinct (Alike Objects)

In how many ways can you permute the letters of the word VIVEKANANDA?

A

$1663200$

B

$3326400$

C

$831600$

D

$4989600$

Question 149: Palette 149/213

Comprehensive Algebra MCQ

Concept: Permutations of Multiset / Like Elements

A train time table must be compiled for various days of the week so that two trains a day depart for three days, one train a day for two days, and three trains a day for two days. How many different time tables can be compiled?

A

$420$

B

$210$

C

$105$

D

$630$

Question 150: Palette 150/213

Comprehensive Algebra MCQ

Concept: Permutations with Repetition and Divisibility Constraints

Find the number of six digit numbers not divisible by 5 that can be formed by using the digits of the number 121202.

A

$50$

B

$30$

C

$40$

D

$60$

Question 151: Palette 151/213

Comprehensive Algebra MCQ

Concept: Complementary Counting on Digits

In a certain country, the numerals in car registration marks range from 1 to 999. Find the number of cases in which the first local car which you see while visiting that country has at least two digits the same in its registration mark.

A

$261$

B

$216$

C

$252$

D

$280$

Question 152: Palette 152/213

Comprehensive Algebra MCQ

Concept: Combinatorial Position Selection and Digit Placement

Find the number of seven digit numbers which have exactly three 9's.

A

$215055$

B

$246080$

C

$198250$

D

$205000$

Question 153: Palette 153/213

Comprehensive Algebra MCQ

Concept: Complementary Counting for Grouped Permutations

In how many ways can 5 identical black balls, 7 identical red balls, and 6 identical green balls be arranged in a row so that at least one ball is separated from balls of the same color?

A

$\frac{18!}{5! 7! 6!} - 6$

B

$\frac{18!}{5! 7! 6!} - 3!$

C

$\frac{18!}{18!}$

D

$\frac{18!}{5! 7! 6!} - 1$

Question 154: Palette 154/213

Comprehensive Algebra MCQ

Concept: Lexicographical Rank of Words with Repeated Letters

A dictionary is printed consisting of 7-lettered words that can be made with the letters of the word CRICKET. If the words are printed in the alphabetic order, as in an ordinary dictionary, find the position of the word CRICKET in that dictionary.

A

$531$

B

$530$

C

$528$

D

$532$

Question 155: Palette 155/213

Comprehensive Algebra MCQ

Concept: Case Analysis and Permutations of Identical Objects in Game Series

Two players A and B play a series of games of chess. The winning player in any game gets 1 point while the losing player gets 0 point. The player who achieves 4 points first, wins the series. If no game ends in a draw, find the number of ways in which the series can be won by A.

A

35

B

28

C

42

D

30

Question 156: Palette 156/213

Comprehensive Algebra MCQ

Concept: Permutations of Objects Not All Distinct (Alike Objects)

In how many ways can you permute the letters of the word CONSTITUTION?

A

$9979200$

B

$19958400$

C

$4989600$

D

$1247400$

Question 157: Palette 157/213

Comprehensive Algebra MCQ

Concept: Permutations with Repeated and Distinct Letters

How many different permutations can be formed from the letters contained in the following words: (a) ZEBRA, (b) BAZAR, (c) SILICIC, (d) ABRACADABRA?

A

(a) 120, (b) 60, (c) 420, (d) 83160

B

(a) 120, (b) 120, (c) 210, (d) 83160

C

(a) 60, (b) 60, (c) 420, (d) 41580

D

(a) 120, (b) 60, (c) 840, (d) 166320

Question 158: Palette 158/213

Comprehensive Algebra MCQ

Concept: Arrangement of Identical Objects in a Line

In how many ways can we plant a line of 10 bulbs consisting of 5 roses, 3 daffodils, and 2 sunflowers?

A

$5040$

B

$2520$

C

$1260$

D

$10080$

Question 159: Palette 159/213

Comprehensive Algebra MCQ

Concept: Sum of Numbers Formed by Permutation of Digits with Repetition

Find the sum of all the numbers that can be formed using all the digits 2, 3, 3, 4, 4, 4.

A

$2222220$

B

$2111100$

C

$2333310$

D

$2444420$

Question 160: Palette 160/213

Comprehensive Algebra MCQ

Concept: Permutations with Specific Subset Selection and Repetitions

There are $4n$ things of which $n$ are alike and all the rest different. Find the number of permutations of $4n$ things taken $2n$ at a time, each permutation containing the $n$ like things.

A

$\frac{(3n)! n!}{(2n)!}$

B

$\frac{(3n)!}{(n!)^2}$

C

$\frac{(3n)!}{n!}$

D

$\frac{(3n)!}{n! (2n)!}$

Question 161: Palette 161/213

Comprehensive Algebra MCQ

Concept: Multinomial Permutation and Unassigned Outcomes

How many ways of tossing 12 dice are there in which each of the values 2, 3, 4, 5, 6 occurs twice?

A

$\frac{12!}{(2!)^5}$

B

$\frac{12!}{(2!)^6}$

C

$12!$

D

$\frac{12!}{2!}$

Question 162: Palette 162/213

Comprehensive Algebra MCQ

Concept: Binomial Sequence Combinations

We toss a coin and assume a head is a success and a tail is a failure. How many trails will lead to 52 successes out of 100 tosses of the coin?

A

$^{100}C_{52}$

B

$^{100}P_{52}$

C

$\frac{100!}{52!}$

D

$2^{100}$

Question 163: Palette 163/213

Comprehensive Algebra MCQ

Concept: Permutations with Repeated Digits and Parity Restrictions

How many seven digits numbers can be formed with the digits 1, 2, 2, 2, 3, 3, 5? How many of them are odd?

A

Total: $420$, Odd: $240$

B

Total: $420$, Odd: $180$

C

Total: $210$, Odd: $120$

D

Total: $840$, Odd: $480$

Question 164: Palette 164/213

Comprehensive Algebra MCQ

Concept: Sum of Geometric Series in Permutations with Repetition

A biologist is studying patterns of male(M) and female(F) children in families. A family type is designated by a code; for example, FMM denotes a family of three children of which the oldest is a female and the other two males. Note that FMM, MFM, and MMF are different types. How many family types are there among families with at least one but not more than seven children?

A

$254$

B

$128$

C

$256$

D

$510$

Question 165: Palette 165/213

Comprehensive Algebra MCQ

Concept: Rank of a Word with Repeated Letters

The letters of the word SURITI are written in all possible orders and are written down as in a dictionary. Find the rank of the word SURITI.

A

$236$

B

$235$

C

$240$

D

$238$

Question 166: Palette 166/213

Comprehensive Algebra MCQ

Concept: Permutations under Restrictions and Complementary Counting

Consider the set $\{a, b, c, d, e\}$. How many three-letter words can be made out of them, with or without meaning? How many of these will have at least one vowel in them? Answer these questions for both cases when repetitions of letters are allowed and when repetitions of letters are not allowed.

A

Repetition allowed: 125 total, 98 with at least one vowel; Repetition not allowed: 60 total, 54 with at least one vowel.

B

Repetition allowed: 125 total, 27 with at least one vowel; Repetition not allowed: 60 total, 6 with at least one vowel.

C

Repetition allowed: 60 total, 54 with at least one vowel; Repetition not allowed: 125 total, 98 with at least one vowel.

D

Repetition allowed: 125 total, 98 with at least one vowel; Repetition not allowed: 60 total, 36 with at least one vowel.

Question 167: Palette 167/213

Comprehensive Algebra MCQ

Concept: Permutations with Position Restrictions

How many different words can be formed with the letters of the word PENCIL when vowels occupy even places?

A

$144$

B

$720$

C

$360$

D

$288$

Question 168: Palette 168/213

Comprehensive Algebra MCQ

Concept: Fixed Position Permutations

How many of the permutations of the word ENGLISH will:
(i) start with E?
(ii) end with H?
(iii) start with E and end with H?

A

(i) $720$, (ii) $720$, (iii) $120$

B

(i) $5040$, (ii) $720$, (iii) $120$

C

(i) $720$, (ii) $720$, (iii) $240$

D

(i) $120$, (ii) $120$, (iii) $720$

Question 169: Palette 169/213

Comprehensive Algebra MCQ

Concept: Sum of Digits at Place Values with Middle Place Restrictions

Three-digit numbers in which the middle digit is a perfect square are formed using the digits 1 to 9. Find the sum of all such numbers.

A

$134055$

B

$124055$

C

$143055$

D

$135045$

Question 170: Palette 170/213

Comprehensive Algebra MCQ

Concept: String Method / Grouping Identical Elements Together

A library has two books each having three copies and three other books each having two copies. In how many ways can all these books be arranged in a shelf so that copies of the same book are not separated?

A

$120$

B

$720$

C

$240$

D

$60$

Question 171: Palette 171/213

Comprehensive Algebra MCQ

Concept: String Method with Internal Permutations

In how many ways can 8 Indians, 4 Americans, and 4 Englishmen be seated in a row so that all persons of the same nationality sit together?

A

$3! \times 8! \times 4! \times 4!$

B

$8! \times 4! \times 4!$

C

$16!$

D

$3! \times 16!$

Question 172: Palette 172/213

Comprehensive Algebra MCQ

Concept: String Method / Grouping Permutations with Repeated Letters

How many different words can be formed with the letters of the word 'UNIVERSITY' so that all the vowels are together?

A

$60480$

B

$120960$

C

$30240$

D

$50400$

Question 173: Palette 173/213

Comprehensive Algebra MCQ

Concept: Relative Order and Block Permutations

Five persons are to address a meeting. If a specified speaker is to speak before another specified speaker, find:
(i) the number of ways in which this could be arranged.
(ii) how many of these arrangements will have the first speaker come immediately before the second?

A

(i) $60$, (ii) $24$

B

(i) $120$, (ii) $24$

C

(i) $60$, (ii) $12$

D

(i) $120$, (ii) $48$

Question 174: Palette 174/213

Comprehensive Algebra MCQ

Concept: Gap and Subgroup Permutation Arrangements

10 IIT and 2 PET students sit in a row. Find the number of ways in which exactly 3 IIT students sit between 2 PET students.

A

$16 \times 10!$

B

$8 \times 10!$

C

$10 \times 10!$

D

$20 \times 10!$

Question 175: Palette 175/213

Comprehensive Algebra MCQ

Concept: Complementary Counting for Unseparated Elements

In a class of 10 students, there are 3 girls. In how many ways can they be arranged in a row such that all the three girls do not sit together?

A

$10! - 8! \times 3!$

B

$10! - 7! \times 3!$

C

$8! \times 3!$

D

$10! - 8!$

Question 176: Palette 176/213

Comprehensive Algebra MCQ

Concept: Non-adjacent Permutations via Complementary Subtraction

Determine the number of permutations of $n$ elements taken all at a time in which two given elements $a$ and $b$ are not adjacent.

A

$(n - 1)! (n - 2)$

B

$(n - 2)! (n - 1)$

C

$n! - (n - 1)!$

D

$(n - 2)! n$

Question 177: Palette 177/213

Comprehensive Algebra MCQ

Concept: Gap Method for Non-Adjacent Permutations

There are 10 candidates for an examination out of which 4 are appearing in Mathematics and the remaining 6 are appearing in different subjects. In how many ways can they be seated in a row so that no two mathematics candidates are together?

A

$604800$

B

$504000$

C

$302400$

D

$725760$

Question 178: Palette 178/213

Comprehensive Algebra MCQ

Concept: Gap Method with Permutations of Subsets

Of the $30!$ permutations of the integers $1, 2, 3, \dots, 30$, how many have the property that multiples of 3 are not in adjacent places (that is, no two of the integers $3, 6, 9, \dots, 27, 30$ are adjacent)?

A

$\frac{20! \times 21!}{11!}$

B

$\frac{30!}{10!}$

C

$\frac{20! \times 21!}{10!}$

D

$\frac{20!}{10!}$

Question 179: Palette 179/213

Comprehensive Algebra MCQ

Concept: Gap Method and Inclusion-Exclusion Principles

In how many ways can the letters AAABBCD be arranged so that:
(i) the two B's are together but no two A's are together?
(ii) no two B's and no two A's are together?

A

(i) 24, (ii) 96

B

(i) 36, (ii) 120

C

(i) 24, (ii) 120

D

(i) 48, (ii) 96

Question 180: Palette 180/213

Comprehensive Algebra MCQ

Concept: Permutations with Repeated Objects and Restrictions

Find the total number of arrangements of the letters of the word INDEPENDENCE. How many of these:
(i) do the words start with P?
(ii) do all the vowels always occur together?
(iii) do all the vowels never occur together?

A

Total: $1663200$; (i) $138600$, (ii) $16800$, (iii) $1646400$

B

Total: $1663200$; (i) $138600$, (ii) $16800$, (iii) $1500000$

C

Total: $3326400$; (i) $277200$, (ii) $33600$, (iii) $3292800$

D

Total: $1663200$; (i) $69300$, (ii) $16800$, (iii) $1646400$

Question 181: Palette 181/213

Comprehensive Algebra MCQ

Concept: To find the number of arrangements where a specific group of items does not sit together, subtract the number of arrangements where they sit together as a single unit from the total number of unrestricted arrangements.

In how many ways can 5 boys and 3 girls be seated in a row so that all the three girls do not sit together?

A

1152

B

36000

C

40320

D

4320

Question 182: Palette 182/213

Comprehensive Algebra MCQ

Concept: When two equal groups of items need to be arranged alternately in a row, there are two distinct starting patterns: either the first group takes the odd positions or the second group takes the odd positions. The total number of arrangements is the sum of the arrangements from both cases.

In how many ways can 4 boys and 4 girls be seated in a row so that boys and girls sit alternate?

A

576

B

1152

C

288

D

2304

Question 183: Palette 183/213

Comprehensive Algebra MCQ

Concept: When the number of items in one group exceeds the other by 1, the group with more items must occupy the odd-numbered positions (1st, 3rd, 5th, etc.) so that no two items of the same group sit adjacent to each other.

In how many ways can 4 boys and 3 girls be seated in a row so that they sit alternately?

A

288

B

144

C

720

D

576

Question 184: Palette 184/213

Comprehensive Algebra MCQ

Concept: For alternating seating arrangements where one category has one more person than the other, there is only one valid seating pattern. The larger group takes all odd spots and the smaller group takes all even spots.

In how many ways can 4 boys and 3 girls be seated in a row so that boys and girls sit alternate?

A

576

B

72

C

144

D

288

Question 185: Palette 185/213

Comprehensive Algebra MCQ

Concept: When two groups have an equal number of members, alternating arrangements can begin with either group. The total number of ways is the sum of arrangements starting with the first group and arrangements starting with the second group.

In how many ways can 4 boys and 4 girls be seated in a row so that boys and girls sit alternate?

A

1152

B

576

C

2304

D

288

Question 186: Palette 186/213

Comprehensive Algebra MCQ

Concept: To form three-letter words with a vowel in the middle, consider each distinct vowel available in the word separately. For each middle vowel, count the number of valid arrangements for the remaining two outer positions, accounting for whether the outer letters are distinct or identical.

How many words of three letters can be formed from the word 'Keppelin', a vowel being always in the middle?

A

63

B

53

C

42

D

31

Question 187: Palette 187/213

Comprehensive Algebra MCQ

Concept: To form a number where specific digits occupy specific positions, find the number of permutations of the odd digits among the odd positions and multiply it by the number of permutations of the even digits among the even positions, accounting for repeated digits.

How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?

A

24

B

18

C

36

D

12

Question 188: Palette 188/213

Comprehensive Algebra MCQ

Concept: First select the person for the steering position from the available steering specialists. Then place the restricted rowers on their designated side, select the remaining rowers required for both sides, and multiply by the permutations of the rowers on each side.

An eight-oared boat is to be manned by a crew chosen from 11 men of whom 3 can steer but cannot row and the rest cannot steer. In how many ways can the crew be arranged if two of the men can only row on the bow side?

A

25920

B

12960

C

51840

D

8640

Question 189: Palette 189/213

Comprehensive Algebra MCQ

Concept: When specific letters must stay together, treat them as a single combined block. To ensure other identical letters remain separated, place all other letters first and then arrange the restricted letters in the available spaces created between them.

How many seven-letter words can be formed by using the letters of the word SUCCESS so that:
(a) the two C are together but no two S are together?
(b) no two C and no two S are together?

A

(a) 36, (b) 72

B

(a) 24, (b) 96

C

(a) 48, (b) 120

D

(a) 12, (b) 108

Question 190: Palette 190/213

Comprehensive Algebra MCQ

Concept: When two specific items must be side by side, treat them as a single block to reduce the total number of items, then multiply by the internal arrangements of the block. To find the arrangements where they are not side by side, subtract the side-by-side arrangements from the total unrestricted arrangements.

In how many ways can 6 books be arranged on a shelf if (a) two particular books must be side by side and (b) if these two books must not be side by side?

A

(a) 120, (b) 600

B

(a) 240, (b) 480

C

(a) 144, (b) 576

D

(a) 240, (b) 720

Question 191: Palette 191/213

Comprehensive Algebra MCQ

Concept: To arrange identical or distinct copies grouped by topic together, treat each topic as a single group. First, arrange the groups, and then multiply by the internal permutations of the books within each group (accounting for identical copies using permutations of multisets).

I have 2 copies of a statistics book, 2 copies of an algebra book, 3 copies of a calculus book, and 1 copy of a book on astronomy. In how many ways can I arrange these books on a shelf so that books of each topic are together?

A

24

B

576

C

144

D

288

Question 192: Palette 192/213

Comprehensive Algebra MCQ

Concept: To keep two specified volumes together in a large set, group them as one unit. To keep two volumes apart, subtract the number of arrangements where they are together from the total factorial of the set size.

A collection of 30 volumes is on a book shelf. How many ways are there of arranging the series (a) for volumes 1 and 2 to be side by side? (b) for volumes 3 and 4 not to be side by side?

A

(a) $2 \times 29!$, (b) $28 \times 29!$

B

(a) $29!$, (b) $30! - 29!$

C

(a) $2 \times 30!$, (b) $29 \times 29!$

D

(a) $29!$, (b) $2 \times 29!$

Question 193: Palette 193/213

Comprehensive Algebra MCQ

Concept: Calculate the total valid 4-digit numbers using the allowed digits (ensuring the thousand's place is non-zero), and subtract the number of 4-digit numbers formed without using the digit 1 at all.

How many different four-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7 so that each number contains one digit 1?

A

1152

B

1080

C

1470

D

2058

Question 194: Palette 194/213

Comprehensive Algebra MCQ

Concept: (i) Fix the first and last letters and arrange the remaining letters accounting for repeated letters.
(ii) Group all vowels as a single unit and arrange the remaining letters and internal vowels.
(iii) Use position analysis for fixed gaps between two specified letters.

In how many ways can the letters of the word PERMUTATIONS be arranged if (i) words start with P and end with S, (ii) vowels are all together, and (iii) there are always 4 letters between P and S?

A

(i) 1814400, (ii) 2419200, (iii) 25401600

B

(i) 3628800, (ii) 1209600, (iii) 12700800

C

(i) 1814400, (ii) 1209600, (iii) 25401600

D

(i) 3628800, (ii) 2419200, (iii) 50803200

Question 195: Palette 195/213

Comprehensive Algebra MCQ

Concept: To find the number of arrangements where two specific items do not appear adjacent to each other, subtract the number of arrangements where they are adjacent from the total number of unrestricted arrangements.

In how many ways can the time-table for Monday be arranged if for this day five lessons are planned: in algebra, geometry, history, geography and literature, provided algebra and geometry do not immediately follow each other?

A

72

B

120

C

48

D

96

Question 196: Palette 196/213

Comprehensive Algebra MCQ

Concept: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Determine the valid choices for the last two places and then fill the remaining positions based on whether repetition is allowed or not.

How many five-digit numbers divisible by 4 can be formed by the digits 1, 2, 3, 4 and 5 if (i) digits can be repeated in the same number? (ii) digits cannot be repeated in the same number?

A

(i) 625, (ii) 24

B

(i) 3125, (ii) 120

C

(i) 750, (ii) 48

D

(i) 1250, (ii) 24

Question 197: Palette 197/213

Comprehensive Algebra MCQ

Concept: A number is divisible by 6 if it is even (ends in 0, 2, or 4) and its sum of digits is divisible by 3. Select 5 distinct digits out of the 6 available digits such that their sum is divisible by 3, and then arrange them ensuring the last digit is even and the first digit is non-zero.

How many five-digit numbers divisible by 6 can be made with the digits 0, 1, 2, 3, 4 and 5 if the digits cannot be repeated in the same number?

A

192

B

216

C

312

D

156

Question 198: Palette 198/213

Comprehensive Algebra MCQ

Concept: A 5-digit telephone number uses the digits 0 through 9, where the first digit can be any digit including 0. When digits are pairwise distinct, the number of ways is given by permutations of 10 available digits taken 5 at a time.

How many five-digit telephone numbers with pairwise distinct digits can be composed?

A

27216

B

30240

C

15120

D

32768

Question 199: Palette 199/213

Comprehensive Algebra MCQ

Concept: A number is divisible by 25 if its last two digits are 25, 50, 75, or 00. Determine the combinations for the last two digits, then count the possibilities for the remaining initial positions without repeating or allowing zero at the leading position depending on repetition rules.

How many numbers of 5 digits divisible by 25 can be made with the digits 0, 1, 2, 3, 4, 5, 6 and 7?

A

2520

B

1050

C

1260

D

840

Question 200: Palette 200/213

Comprehensive Algebra MCQ

Concept: To count arrangements where multiple specific pairs of items are restricted from being together, apply the Principle of Inclusion-Exclusion. Subtract the arrangements where at least one pair is together from the total permutations, and add back the arrangements where both pairs are simultaneously together.

Find the number of arrangements of the letters $a, b, c, d$ in which neither $a, b$ nor $c, d$ come together.

A

8

B

10

C

12

D

16

Question 201: Palette 201/213

Comprehensive Algebra MCQ

Concept: Assign the restricted crew members to their required sides first. Then select from the remaining unrestricted crew members to complete each side, and multiply by the permutations of arranging the rowers on both the bow and stroke sides.

A boat is to be manned by eight men, of whom 2 can only row on bow side and 1 can only row on stroke side. In how many ways can the crew be arranged?

A

5760

B

1440

C

2880

D

4320

Question 202: Palette 202/213

Comprehensive Algebra MCQ

Concept: To arrange items of different categories such that members of each category sit together, treat each nationality group as a single unit. First find the arrangements of the groups, then multiply by the internal permutations of each individual group.

In a dinner party there are 10 Indians, 5 Americans and 5 Englishmen. In how many ways can they be arranged in a row so that all persons of the same nationality sit together?

A

$10! \times 5! \times 5!$

B

$3! \times 10! \times 5! \times 5!$

C

$6 \times 10! \times 25!$

D

$3! \times 20!$

Question 203: Palette 203/213

Comprehensive Algebra MCQ

Concept: To arrange items such that two specific items are not adjacent (successive), calculate the total unrestricted arrangements of all items and subtract the number of arrangements where the two specified items are placed together as a single unit.

Six papers are set in an examination, 2 of them in mathematics. In how many different orders can the papers be given if two mathematics papers are not successive?

A

240

B

360

C

480

D

600

Question 204: Palette 204/213

Comprehensive Algebra MCQ

Concept: When specific identical letters must not be adjacent, use the gap method: arrange all other letters first, then insert the restricted identical letters into the spaces created between and at the ends of the arranged letters.

In how many ways can the letters of the word 'PLANTAIN' be arranged so that the two 'A's do not come together?

A

7560

B

2520

C

3780

D

1260

Question 205: Palette 205/213

Comprehensive Algebra MCQ

Concept: When items from two groups must alternate and both groups have an equal number of members, determine the distinct starting configurations (consonant first or vowel first) and multiply by the permutations of each group while accounting for duplicate letters.

In how many different ways can the letters of the word 'SALOON' be arranged if the consonants and vowels must occupy alternate places?

A

36

B

18

C

72

D

24

Question 206: Palette 206/213

Comprehensive Algebra MCQ

Concept: Calculate total permutations of a multiset of letters. Subtract the arrangements where specified letters are grouped together to find non-adjacent cases. For conditional positions, fix the letters at the specified endpoints and arrange the rest.

How many words can be formed by using the letters of the word 'BHARAT'? How many of these words will not contain B and H together? How many of these start with B and end with T?

A

Total: 360; Without BH together: 240; Starting with B and ending with T: 12

B

Total: 360; Without BH together: 120; Starting with B and ending with T: 24

C

Total: 720; Without BH together: 480; Starting with B and ending with T: 12

D

Total: 360; Without BH together: 240; Starting with B and ending with T: 24

Question 207: Palette 207/213

Comprehensive Algebra MCQ

Concept: Use basic factorials for unrestricted permutations, block arrangements for grouped items, gap and unit concepts for partial restrictions, and position-matching for alternating arrangements.

There are nine different books on a shelf; four are red and five are green. In how many different orders is it possible to arrange the books on the shelf if:
(a) there are no restrictions;
(b) the red books must be together and the green books together;
(c) the red books must be together whereas the green books may be, but need not be, together;
(d) the colours must alternate, i.e., no two books of the same colour may be adjacent?

A

(a) 362880, (b) 5760, (c) 17280, (d) 2880

B

(a) 362880, (b) 2880, (c) 14400, (d) 1440

C

(a) 362880, (b) 5760, (c) 28800, (d) 2880

D

(a) 181440, (b) 2880, (c) 17280, (d) 1440

Question 208: Palette 208/213

Comprehensive Algebra MCQ

A straight is a five-card hand containing consecutive values. Find (i) how many different straights there are, and (ii) how many straights there are if the cards are not all from the same suit.

A

(i) $9180$, (ii) $9216$

B

(i) $9216$, (ii) $9180$

C

(i) $8192$, (ii) $8156$

D

(i) $9216$, (ii) $9200$

Question 209: Palette 209/213

Comprehensive Algebra MCQ

$n$ different objects are arranged in a row. In how many ways can $3$ objects be selected so that (i) all three objects are consecutive, and (ii) all three objects are not consecutive?

A

(i) $n - 2$, (ii) $\frac{(n - 2)(n^2 - n - 6)}{6}$

B

(i) $n - 3$, (ii) $\frac{n(n - 1)(n - 2)}{6}$

C

(i) $n - 2$, (ii) $\frac{n(n - 2)(n - 4)}{6}$

D

(i) $n - 1$, (ii) $\frac{(n - 1)(n - 2)(n - 3)}{6}$

Question 210: Palette 210/213

Comprehensive Algebra MCQ

There are $n$ intermediate stations on a railway line from one terminus to another. In how many ways can the train stop at $3$ of these intermediate stations if (i) all three stations are consecutive, and (ii) at least two of the stations are consecutive?

A

(i) $n - 2$, (ii) $(n - 2)^2$

B

(i) $n - 3$, (ii) $(n - 1)(n - 2)$

C

(i) $n - 2$, (ii) $n(n - 2)$

D

(i) $n - 1$, (ii) $(n - 2)^3$

Question 211: Palette 211/213

Comprehensive Algebra MCQ

Concept: Selection of vertices in a polygon to form triangles with or without common sides.

Triangles are formed from the vertices of an $n$-sided polygon. Find the number of triangles so that (i) at least one side of the triangle coincides with the side of the polygon, and (ii) no side of the triangle coincides with the side of the polygon.

A

(i) $n(n - 3)$, (ii) $\frac{n(n - 4)(n - 5)}{6}$

B

(i) $n(n - 2)$, (ii) $\frac{n(n - 1)(n - 2)}{6}$

C

(i) $\frac{n(n - 3)}{2}$, (ii) $\frac{(n - 3)(n^2 - 4)}{6}$

D

(i) $n(n - 4)$, (ii) $\frac{n(n - 3)(n - 4)}{6}$

Question 212: Palette 212/213

Comprehensive Algebra MCQ

Concept: Selection of non-adjacent items in a circular arrangement.

There are eight stations on a train route in a circular order. Find the number of selections of $3$ stations such that no two stations are adjacent.

A

$20$

B

$16$

C

$24$

D

$12$

Question 213: Palette 213/213

Comprehensive Algebra MCQ

Concept: Permutations with forbidden adjacent elements (complementary counting).

How many $7$-digit numbers are there such that the digits are distinct integers taken from the set $S = \{1, 2, \dots, 9\}$ and such that the integers $5$ and $6$ do not appear consecutively in either order?

A

$151200$

B

$181440$

C

$30240$

D

$152100$