CAT Permutation and Combination Questions​ with Answers PDF Download

Boost your CAT preparation with 30 permutation and combination CAT questions with answers. Read on to know essential concepts in Permutations and Combinations in CAT 2026! 

Preparing for the CAT exam can feel like a big challenge, especially when it comes to the Quantitative Aptitude section. Among the many topics, Probability, Permutations, and Combinations (P&C) are crucial. These subjects often appear in the exam. They test how you think and solve problems. 

Here we will help you understand these important concepts. It will show you how to tackle tough Permutation and Combination CAT Questions using smart strategies.

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What Are Permutations in CAT Exam 2026?

Permutations in the CAT Syllabus deal with arranging objects, numbers, or people in a particular order. Here, the order of the items matters, so changing their position creates a different arrangement.

Permutation questions are usually covered under the Quantitative Aptitude section. These questions ask you to find the number of possible arrangements for a given set of items.

What Are Combinations in CAT Exam 2026?

Combinations are selections where order does not matter. Choosing 3 team members from a group of 7 is a combination problem. The set {A, B, C} is the same as {C, A, B}. This difference is key in permutation and combination CAT questions because it affects both the method and the count.

The Role of Factorials in P&C

Factorial is the building block for counting. It multiplies down from a number to 1.

n! = n × (n − 1) × (n − 2) × … × 1

For example, 5! = 120. Many formulas in permutations and combinations use factorials, so get comfortable reducing them.

CAT Permutation and Combination Resources

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Important Permutation and Combination CAT Questions 2026

Here are 30 most important Permutation and combination CAT questions to practice:

1.  After two successive increments, Gopal's salary became 187.5% of his initial salary. If the percentage of salary increase in the second increment was twice that of the first increment, then the percentage of salary increase in the first increment was

1. 30
2. 25
3. 27.5
4. 20

Answer: Option 2

2. The number of all natural numbers up to 1000 with non-repeating digits is:

1. 648
2. 585
3. 738
4. 504

Answer: Option 3

3. The number of integers greater than 2000 that can be formed with the digits 0, 1, 2, 3, 4, 5, using each digit at most once, is

1. 1200
2. 1420
3. 1440
4. 1480

Answer: Option 3

4. The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

1. 2222
2. 2592
3. 2442
4. 3333

Answer: Option 1

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5. The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive integers such that x ≤ 40, y ≤ 12, and z ≤ 12 is

1. 101
2. 99
3. 87
4. 105

Answer: Option 2

6. In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

1. 16
2. 20
3. 14
4. 15

Answer: Option 1

7. In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens?

Answer: 6

8. How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

1. 499
2. 500
3. 375
4. 376
5. 501

Answer: Option 4

Read more: CAT Ratio and Proportion Questions

9. What is the number of distinct terms in the expansion of (a + b + c)20?

1. 231
2. 253
3. 242
4. 210
5. 228

Answer: Option 1

10. For general n, how many enemies will each member of S have?

1. n – 3
2. 12(n2-3n-2)
3. 2n - 7
4. 12(n2-5n+6)
5. 12(n2-7n+14)

Answer: Option 4

11. In a tournament, there are n teams T1 , T2 ....., Tn with n > 5. Each team consists of k players, k > 3. The following pairs of teams have one player in common:m T1 & T2 , T2 & T3 ,......, Tn − 1 & Tn , and Tn & T1. No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together?

1. n(k – 1)
2. k(n – 1)
3. n(k – 2)
4. k(k – 2)
5. (n – 1)(k – 1)

Answer: Option 1

Check: How to prepare for Quantitative Aptitude for CAT exam?

12. There are 6 tasks and 6 people. Task 1 cannot be assigned to either person 1 or person 2; task 2 must be assigned to either person 3 or person 4. Every person is to be assigned one task. In how many ways can the assignment be done?

1. 144
2. 180
3. 192
4. 360
5. 716

Answer: Option 1

13. In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls, and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is

1. 200
2. 216
3. 235
4. 256

Answer: Option 1

14. Let S be the set of five-digit numbers formed by the digits 1, 2, 3, 4 and 5, using each digit exactly once, such that exactly two odd positions are occupied by odd digits. What is the sum of the digits in the rightmost position of the numbers in S?

1. 228
2. 216
3. 294
4. 192

Answer: Option 2

15. N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?

1. 5
2. 7
3. 9
4. None of the above

Answer: Option 2

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16. A new flag is to be designed with six vertical stripes using some or all of the colours yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent stripes have the same colour is:

1. 12 × 81
2. 16 × 192
3. 20 × 125
4. 24 × 216

Answer: Option 1

17. Twenty-seven people attend a party. Which one of the following statements can never be true?

1. There is a person at the party who is acquainted with all twenty-six others.
2. Each person in the party has a different number of acquaintances.
3. There is a person in the party who has an odd number of acquaintances.
4. In the party, there is no set of three mutual acquaintances.

Answer: Option 2

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18. How many three-digit positive integers, with digits x, y and z in the hundreds, ten, and units place respectively, exist such that x < y, z < y and x ≠ 0?

1. 245
2. 285
3. 240
4. 320

Answer: Option: 3

19. There are 6 boxes numbered 1, 2,..., 6. Each box is to be filled with either a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is

1. 5
2. 21
3. 33
4. 60

Answer: Option 2

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20. A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any other point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition

1. 11 ≤ e ≤ 66
2. 10 ≤ e ≤ 66
3. 11 ≤ e ≤ 65
4. 0 ≤ e ≤ 11

Answer: Option 1

21. There are 12 towns grouped into four zones, with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required?

1. 72
2. 90
3. 96
4. 144

Answer: Option 2

22. An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, ... , 9 such that the first digit of the code is non-zero. The code, handwritten on a slip, can however potentially create confusion, when read upside down – for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?

1. 80
2. 78
3. 71
4. 69

Answer: Option 3

23. n1, n2, n3 ... n10 are 10 numbers such that n1 > 0 and the numbers are given in ascending order. How many triplets can be formed using these numbers such that in each triplet, the first number is less than the second number, and the second number is less than the third number?

1. 109
2. 27
3. 36
4. None of these

Answer: Option 4

24. How many numbers between 0 and one million can be formed using 0, 7 and 8?

1. 486
2. 1086
3. 728
4. None of these

Answer: Option 3

25 In how many ways can we choose a black and a white square on a chessboard such that the two are not in the same row or column?

1. 32
2. 96
3. 24
4. None of these

Answer: Option 4

26. How many four-letter passwords can be formed by using symmetrical letters only? (Repetitions not allowed)

1. 1086
2. 255
3. 7920
4. None of these

Answer: Option 3

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27. How many three-lettered words can be formed such that at least one symmetrical letter is there?

1. 12870
2. 18330
3. 16420
4. None of these

Answer: Option 4

28. Let n be the number of different 5-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?

1. 144
2. 168
3. 192
4. None of these

Answer: Option 3

29. One red flag, three white flags and two blue flags are arranged in a line such that no two adjacent flags are of the same colour. The flags at the two ends of the line are of different colours. In how many different ways can the flags be arranged?

1. 6
2. 4
3. 10
4. 2

Answer: Option 1

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30. Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number nine appears once. If Sam were to use a trial-and-error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?

1. 1000
2. 2430
3. 3402
4. 3006

Answer: Option 3

Important Fundamental Formulas for CAT 2026

The Permutation Formula: P(n, r)

The number of ways to arrange r items chosen from n distinct items is:

P(n, r) = n! / (n − r)!

Example, arrange 3 books out of 5 on a shelf:

• Choose 3 positions.
• Place one of 5 books, then one of 4, then one of 3.
• That gives 5 × 4 × 3, which matches P(5, 3) = 60.

Steps to compute:

• Write n × (n − 1) × … up to r terms.
• Stop once you have r factors.
• Multiply to get the count.

The Combination Formula: C(n, r)

The number of ways to select r items from n distinct items, order ignored, is:

C(n, r) = n! / (r! × (n − r)!)

Example, choose 2 fruits from 4 types: C(4, 2) = 6.

Use symmetry to save time: C(n, r) = C(n, n − r).

How to Solve Permutation and Combination CAT Questions 2026?

Permutation and Combination CAT questions test your logical thinking and counting skills under constraints. To solve them efficiently, follow these key strategies:

1. Start with Concepts: Learn the core principles of permutations and combinations, focusing on when to use each.

2. Recognize Patterns: Understand common CAT question formats like circular arrangements, selection under restrictions, and distribution problems.

3. Apply Constraints Carefully: Pay close attention to wording conditions like “at least,” “together,” or “not together” often change the approach.

4. Use Smart Counting: When direct counting is hard, use techniques like complementary counting or fixing one element to simplify.

5. Practice Strategically: Work on a mix of basic and CAT Permutation and Combination Questions​ to build both conceptual clarity and exam-specific intuition.

Conclusion

Mastering Probability, Permutation and Combination CAT Questions is a key step towards acing the CAT Quantitative Aptitude section.

First, make sure your basics are clear. Go through the important definitions, formulas, and question types before moving on to tougher questions. Solving CAT previous year papers and taking mock tests regularly will also help you get used to the kind of questions asked in CAT.

Do not try to memorise every formula without knowing where to use it. Keep a formula sheet for quick revision and note down the questions you get wrong during practice. Going back to these mistakes and revising them can make a noticeable difference in your preparation.