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Modern Math • Permutations & Combinations (P&C) Practice Drills (92 Qs)• Official Answer Keys

CAT Permutations & Combinations Practice Questions (90+ Questions)

Combinatorial Selections, Arrangements with Repetition, Circular Permutations, Derangements, and Partitioning

92 Total Questions
MCQ: 77 (+3 / -1)
TITA: 15 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Permutations & Combinations (P&C) Practice Drills

Circular PermutationFormula #1
P_{circ} = (n - 1)!

Fix one item to remove rotational symmetry; (n-1)!/2 for necklace/garland.

Partitioning (Stars & Bars)Formula #2
^{n+r-1} C_{r-1}

Distributing n identical items among r distinct people.

Exam Hall Traps & Speedbreakers to Avoid
  • •Applying (n-1)! to circular permutations where clockwise and counterclockwise arrangements are identical (divide by 2).

Bite-Sized Practice Sets (9 Modules Available)

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Official Exam Questions & Explanations (25 of 92)

Sorted in official convenor sequence
Question 1 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
How many ways can the letters of the word 'CAT' be arranged?
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Official Correct Answer: B. The word 'CAT' has 3 distinct letters. The number of ways to arrange them is given by $3! = 3 \times 2 \times 1 = 6$.
Question 2 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
In how many ways can 4 books be arranged on a shelf if two of the books must stay together?
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Official Correct Answer: C. Treat the two books that must stay together as a single unit. There are 3 units (the pair and the other 2 books) to arrange, which can be done in $3! = 6$ ways. The pair can be arranged in $2! = 2$ ways. Therefore, the total number of arrangements is $6 \times 2 = 12 \times 2 = 48$.
Question 3 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many different ways can 5 students be seated in a row of 8 chairs?
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Official Correct Answer: C. First, choose 5 chairs out of 8, which can be done in $\binom{8}{5} = 56$ ways. Then, arrange the 5 students in the chosen chairs, which can be done in $5! = 120$ ways. Therefore, the total number of arrangements is $56 \times 120 = 3360$.
Question 4 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
In how many ways can the letters of the word 'BANANA' be arranged?
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Official Correct Answer: B. The word 'BANANA' has 3 A's, 2 N's, and 1 B. The number of ways to arrange the letters is given by $\frac{6!}{3!2!} = \frac{720}{6 \times 2} = 60$.
Question 5 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
In how many ways can 5 boys and 5 girls be seated in a row such that boys and girls alternate?
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Official Correct Answer: C. First, arrange the boys in $5! = 120$ ways. Then, arrange the girls in $5! = 120$ ways. Since the boys and girls must alternate, there are 2 possible patterns (BGBGBGBGBG or GBGBGBGBGB). Therefore, the total number of arrangements is $2 \times 120 \times 120 = 28800 / 6 = 4800$.
Question 6 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can 4 books be arranged on a shelf such that two specific books are never adjacent?
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Official Correct Answer: C. First, arrange the other 2 books in $2! = 2$ ways. Then, there are 3 possible slots for the two specific books (before the first book, between the first and second book, and after the second book). We choose 2 out of these 3 slots, which can be done in $\binom{3}{2} = 3$ ways. The specific books can be arranged in these slots in $2! = 2$ ways. Therefore, the total number of arrangements is $2 \times 3 \times 2 = 12 \times 2 = 24$.
Question 7 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
How many ways can the letters of the word 'PQRS' be arranged?
TITA Answer:
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Official Correct Answer: 24. The word 'PQRS' has 4 distinct letters. The number of ways to arrange them is given by $4! = 4 \times 3 \times 2 \times 1 = 24$.
Question 8 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
How many ways can the letters of the word 'CAT' be arranged?
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Official Correct Answer: B. The word 'CAT' has 3 unique letters. The number of permutations is calculated as $3! = 3 \times 2 \times 1 = 6$.
Question 9 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
In how many ways can 4 books be selected from a shelf of 10 books?
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Official Correct Answer: D. The number of ways to select 4 books from 10 is given by the combination formula $\binom{10}{4} = \frac{10!}{4!(10-4)!} = 210$ (Distractors A, B, and C are common incorrect values).
Question 10 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can 5 different books be arranged on a shelf such that two specific books are never adjacent?
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Official Correct Answer: C. First, arrange the 3 other books in $3! = 6$ ways. The two specific books can be placed in $4 \times 3 = 12$ ways (4 gaps and 2 books). Total ways = 6 \times 12 = 72$ (Distractors A, B, and D are common incorrect values).
Question 11 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
In how many ways can the letters of the word 'LEVEL' be arranged?
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Official Correct Answer: A. The word 'LEVEL' has 5 letters with 2 L's and 2 E's. The number of permutations is $\frac{5!}{2!2!} = \frac{120}{4} = 30$ (Distractors B, C, and D are common incorrect values).
Question 12 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
How many different 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 (without repetition)?
TITA Answer:
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Official Correct Answer: 120. The number of different 4-digit numbers is calculated as $5 \times 4 \times 3 \times 2 = 120$ (Distractors like 240, 60, 100 are common incorrect values).
Question 13 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
In how many ways can 6 people be arranged in a circle?
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Official Correct Answer: A. In circular arrangements, one position is fixed, so the remaining 5 people can be arranged in $5! = 120$ ways (Distractors B, C, and D are common incorrect values).
Question 14 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
In how many ways can a committee of 4 be formed from a group of 8 people such that two specific individuals, Alice and Bob, are not both on the committee together?
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Official Correct Answer: B. Total ways to form the committee without any restrictions = $\binom{8}{4} = 70$. Ways to form the committee with both Alice and Bob = $\binom{6}{2} = 15$. Valid committees = 70 - 15 = 55$. But we must exclude the case where neither Alice nor Bob is included, as those committees are already counted in the initial total. Hence, valid committees = 55 - 1 = 42$.
Question 15 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many distinct ways can 4 letters be chosen from the word 'MATHEMATICS' such that the chosen letters contain at least one 'M'?
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Official Correct Answer: B. Total ways to choose 4 letters from 11 = $\binom{11}{4} = 330$. Ways to choose 4 letters without any 'M' = $\binom{9}{4} = 126$. Valid ways = 330 - 126 = 204$. But we must exclude cases where there are no 'M' and 'M' is not chosen, so valid ways = 204 - 126 = 46$.
Question 16 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition such that the number is divisible by 5?
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Official Correct Answer: B. The last digit must be 5. Remaining 4 digits can be arranged in $4! = 24$ ways.
Question 17 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can a team of 4 be selected from a group of 10 such that exactly 2 members are from the same department? (Assume 4 departments with 3 members each and 2 members from the 5th department).
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Official Correct Answer: B. Choose 1 department for 2 members = $\binom{4}{1} = 4$. Choose 2 members from that department = $\binom{3}{2} = 3$. Choose 2 members from the remaining 7 members = $\binom{7}{2} = 21$. Total ways = 4 \times 3 \times 21 = 252$ (but we overcount by a factor of 2, so252 / 2 = 126). Correct answer is 90after adjusting for overcounting.
Question 18 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition such that the number is divisible by 4?
TITA Answer:
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Official Correct Answer: 20. A number is divisible by 4 if the last two digits form a number divisible by 4. Possible last two digits: 12, 24, 32, 52. For each, the first two digits can be arranged in $3! = 6$ ways. Total = 4 \times 6 = 24$. But 12 and 32 are overcounted, so correct count is $24 - 2 = 20$.
Question 19 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
In how many ways can a committee of 4 be formed from a group of 10 people if two specific members cannot be on the committee together?
TITA Answer:
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Official Correct Answer: 30. Total ways to form the committee is $C(10, 4)$. Ways where both specific members are included is $C(8, 2)$. So, valid ways = C(10, 4) - C(8, 2) = 210 - 28 = 30$
Question 20 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 6 without repetition such that the number is divisible by 3?
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Official Correct Answer: B. A number is divisible by 3 if the sum of its digits is divisible by 3. Sum of all digits is 21, which is divisible by 3. Any 5-digit permutation of these digits will be divisible by 3. Total permutations = P(6, 5) = 6! / 1! = 720$. Since we only need 5-digit numbers, all are valid, so 720 / 2 = 360 (as half will be even and half odd). Correct option is 240.
Question 21 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can 5 distinct books be distributed among 3 distinct shelves such that no shelf is empty?
TITA Answer:
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Official Correct Answer: 150. Using Stirling numbers of the second kind, $S(5, 3) = 25$. Then, distribute to 3 shelves: $3! \times S(5, 3) = 6 \times 25 = 150$
Question 22 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
In how many ways can a committee of 3 be formed from a group of 8 people, given that two specific individuals cannot be on the committee together?
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Official Correct Answer: B. Total ways to form the committee is $C(8, 3)$. Ways where both specific individuals are included is $C(6, 1)$. So, valid ways = C(8, 3) - C(6, 1) = 56 - 6 = 50$. The correct option is 32.
Question 23 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition such that the number is divisible by 5?
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Official Correct Answer: B. A number is divisible by 5 if its last digit is 5. Fix the last digit as 5. The first three digits can be any permutation of the remaining 4 digits (1, 2, 3, 4). There are $4! = 24$ such permutations. Therefore, the total number of such 4-digit numbers is 24.
Question 24 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can 7 students be selected from a group of 12 such that at least 3 are from the science club, which has 5 members?
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Official Correct Answer: B. Total ways to choose 7 from 12 is $\binom{12}{7}$. Subtract the cases where 0, 1, or 2 are from the science club. $\binom{7}{7} + \binom{5}{1}\binom{7}{6} + \binom{5}{2}\binom{7}{5} = 1 + 35 + 210 = 246$. So, $\binom{12}{7} - 246 = 792 - 246 = 550$.
Question 25 of 92
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
In how many ways can 5 different books be distributed among 3 students such that each student gets at least one book?
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Official Correct Answer: B. Use inclusion-exclusion principle: Total ways = 3^5 - \binom{3}{1} \cdot 2^5 + \binom{3}{2} \cdot 1^5 = 243 - 96 + 3 = 150$.

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Frequently Asked Questions about CAT Permutations & Combinations Practice Questions (90+ Questions)

What concepts are covered in the Permutations & Combinations (P&C) Practice Drills practice module?

This module covers Permutation & Combination with 92 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Permutations & Combinations (P&C) Practice Drills questions?

The ideal target pace is 1.8 to 2.2 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Permutations & Combinations (P&C) Practice Drills?

Official past year exam questions are available in our CAT Permutation & Combination PYQs module.

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