← Back to Mathematics-II

Permutations and Combinations

214 Questions MCQ (Single Correct) Start Comprehensive Algebra Test
All Exams Comprehensive Algebra
Q1 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q2 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q3 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q4 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q5 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
Example 5
Q6 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q7 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q8 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q9 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q10 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q11 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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
Q12 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$240$
Q13 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$48$
Q14 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$256$
Q15 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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 $4096$
Q16 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$1000000$
Q17 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$8 \times 9^{n-1}$
Q18 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$10$
Q19 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$810$
Q20 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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) $800$
Q21 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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)
Q22 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$11$
Q23 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$60$
Q24 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$128$
Q25 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$8280$
Q26 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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 $980$
Q27 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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 $768$
Q28 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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 $100$
Q29 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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.
$240$
Q30 Comprehensive Algebra 1. Principle of Counting MCQ
26 Jul 2026
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) $6$
Q31 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$120$
Q32 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$35$
Q33 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$125$
Q34 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$240$
Q35 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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 $48$
Q36 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$60$
Q37 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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 $k = 6$
Q38 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$36$
Q39 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$60$
Q40 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$18$
Q41 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$240$
Q42 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$120$
Q43 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$600$
Q44 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$200$
Q45 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$45$
Q46 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$656$
Q47 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$224$
Q48 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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) $2^{2n} \times n!$
Q49 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$240$
Q50 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$120$