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Permutations and Combinations

214 Questions Start Comprehensive Algebra Test
All Exams Comprehensive Algebra
Q51 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$15120$
Q52 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$216$
Q53 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$48$
Q54 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$729$
Q55 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$7$
Q56 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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 $12$
Q57 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$1365$
Q58 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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 $15120$
Q59 Comprehensive Algebra Permutations MCQ
26 Jul 2026
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.
$\frac{k!}{n!(k - n)!}$
Q60 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$24$
Q61 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$4000$
Q62 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$64$
Q63 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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) $60$
Q64 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$608400$
Q65 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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) $150$
Q66 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$n!$
Q67 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$2743$
Q68 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$3! \times 12!$
Q69 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$6! \times 5!$
Q70 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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 $4! \times 3^9$
Q71 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
Q72 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$11111100$
Q73 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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) $7716$
Q74 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$256$
Q75 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$3800000$
Q76 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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) $1200$
Q77 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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) $15600000$
Q78 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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.
$360$
Q79 Comprehensive Algebra Permutations with Repetition MCQ
26 Jul 2026
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) $4088$
Q80 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$96$
Q81 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$32448$
Q82 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$26664$
Q83 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$3888860$
Q84 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
Q85 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$720$
Q86 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$\frac{13}{6}$
Q87 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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
Q88 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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
Q89 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$10^6$
Q90 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$4 \times 3^6$
Q91 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$9 \times 9^4$
Q92 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$5 \times 10^4$
Q93 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$9 \times 10^4$
Q94 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
Concept: Permutations and Counting Principles
Find the number of positive integers with distinct digits.
A.
$8877690$
B.
$9876540$
C.
$8123450$
D.
$7654320$
Q95 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$725760$
Q96 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$41112$
Q97 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$4999960$
Q98 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$62220$
Q99 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$608$
Q100 Comprehensive Algebra Permutations Sum & Rank MCQ
26 Jul 2026
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.
$360, 60, 180$