If you’re preparing for CAT 2026, permutation and combination is one of the important Quantitative Aptitude topics to understand. A CAT 2026 permutation and combination question may look easy at first, but one small counting mistake can double your answer. You might count every arrangement twice without even realising it.
That one slip explains why permutation and combination has a harder reputation than it deserves. The maths is simple. The trouble is knowing which rule to apply.
If you searched for CAT 2026 permutation and combination, you likely want three things: the formulas that matter, faster ways to solve questions, and problems to practise with answers. This guide covers all three. You can also download a formula-and-practice PDF further down.
Short answer: Use a permutation (nPr) when the order of the chosen items matters, and a combination (nCr) when it does not. In CAT, most P&C questions add conditions such as “always together”, “never together” or “at least one”. Solve those with the glue method, the gap method or the complement method.
Where Permutation and Combination Fits in CAT 2026
| Detail | Information |
|---|---|
| CAT 2026 exam date | 29 November 2026 (Sunday) |
| Conducting institute | IIM Indore |
| QA section | 22 questions, 40 minutes (as in recent years) |
| Question types | MCQ and TITA (type in the answer) |
| Marking | +3 for a correct answer, −1 for a wrong MCQ, no negative marking for TITA |
| Where P&C sits | Modern Maths, along with Probability and Set Theory |
| Weightage | Low, and it changes from year to year |
Source: iimcat.ac.in
Given the low weightage, why bother? Two reasons. A solid grip on counting makes Probability much easier, and it sharpens case-by-case thinking that helps in some logical reasoning sets. A P&C question is also one of the few QA questions where a clean method can finish the job in under a minute.
You do not need to study everything in this topic. The next sections cover what matters most.
Download the CAT 2026 Permutation and Combination PDF
The PDF puts every formula, the seven tricks and the ten practice questions with the answer key on two pages. Print it and keep it beside your notebook.
Permutation vs Combination: The 10-Second Test
Ask one question before you touch a formula: if I swap two chosen items, does the outcome change?
- Yes → the order matters → use a permutation.
- No → the order does not matter → use a combination.
Take 3 people from a group of 8:
- Choosing a team of 3 gives ⁸C₃ = 56 ways. The team {A, B, C} is the same team in any order.
- Choosing a president, vice-president and secretary gives ⁸P₃ = 336 ways. Here A as president and B as VP differs from B as president and A as VP.
The two answers differ by exactly 3! = 6. That relation, nPr = nCr × r!, is a useful way to check your work.
CAT 2026 Permutation and Combination Formulas
| Concept | Formula or rule |
|---|---|
| Fundamental principle | Independent steps: multiply (m × n). Either-or cases: add (m + n) |
| Factorial | n! = n × (n−1) × … × 1, and 0! = 1 |
| Permutation | ⁿPᵣ = n! / (n−r)! |
| Combination | ⁿCᵣ = n! / [r!(n−r)!] |
| Useful identities | ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎; ⁿCᵣ + ⁿC₍ᵣ₋₁₎ = ⁿ⁺¹Cᵣ |
| Repetition allowed | r places, n choices each: nʳ |
| Items with repeats | n! / (p! q! r!) for p, q, r identical items of each kind |
| Circular arrangement | (n−1)! |
| Necklace or garland | (n−1)!/2 |
Selection, Grouping and Distribution Formulas
| Situation | Formula |
|---|---|
| Divide n distinct items into unequal groups of a, b, c | n! / (a! b! c!) |
| Divide n distinct items into k equal groups of size a | n! / [(a!)ᵏ · k!] |
| n identical items into r distinct boxes (empty boxes allowed) | ⁿ⁺ʳ⁻¹Cᵣ₋₁ |
| n identical items into r distinct boxes (each box gets at least one) | ⁿ⁻¹Cᵣ₋₁ |
| n distinct items into r distinct boxes, none empty | rⁿ − ʳC₁(r−1)ⁿ + ʳC₂(r−2)ⁿ − … |
| Derangements (no item in its own place) | D₃ = 2, D₄ = 9, D₅ = 44 |
| Diagonals of an n-sided polygon | n(n−3)/2 |
| Triangles from n points, k of them collinear | ⁿC₃ − ᵏC₃ |
| Rectangles in an m × n grid of squares | ⁽ᵐ⁺¹⁾C₂ × ⁽ⁿ⁺¹⁾C₂ |
| Paths with a steps right and b steps up | ⁽ᵃ⁺ᵇ⁾Cₐ |
For the theory behind these, the NCERT Class 11 chapter on Permutations and Combinations, hosted by IIT Kanpur’s SATHEE platform, builds the fundamentals from scratch.
7 Tricks to Solve CAT Permutation and Combination Questions Faster
These tricks target the counting logic. For multiplication and calculation shortcuts, see our guide on CAT Quant shortcuts and tricks.
1. Run the order test first
Decide between nPr and nCr before anything else. Most wrong answers start here.
2. Use the glue method for “always together”
Treat the items that must stay together as one block. Arrange the blocks, then arrange the items inside the block.
3. Use the gap method for “never together”
Arrange the other items first. Then drop the restricted items into the gaps. With n items in a row, you get n + 1 gaps.
4. Use the complement method for “at least”
Count the total, then subtract the unwanted case. “At least one woman” becomes total − no women.
5. Divide by k! for a fixed relative order
If k specific items must appear in one fixed order, divide the total arrangements by k!. Seven people in a row, with A somewhere before B, gives 7!/2 = 2,520 ways.
6. Use stars and bars for identical items
Identical chocolates into distinct boxes is a stars-and-bars problem. If every box needs at least one, hand each box one item first. Then distribute the rest freely.
7. Fill the most restricted place first
In number-formation questions, start with the place that has the strictest rule. That is usually “no zero in front” or “last digit even”.
Watch the limits. The glue method gives the wrong count when the block’s internal order is also restricted. The gap method fails if the restricted items are not all of one type. Check the condition before you commit to a trick.
Solved Examples: CAT-Level Permutation and Combination Questions
Example 1: Number formation
How many 4-digit even numbers can you form from the digits 0 to 6 if no digit repeats?
Solution: Split by the last digit.
- Last digit 0: the other three places take 6 × 5 × 4 = 120.
- Last digit 2, 4 or 6 (3 choices): the first digit cannot be 0 or the last digit, so it has 5 choices. The middle two places then take 5 × 4 = 20. That gives 3 × 5 × 20 = 300.
Total = 120 + 300 = 420.
Example 2: Glue method
In how many ways can you arrange the letters of ADMISSION so that all the vowels stay together?
Solution: The vowels A, I, I, O form one block. The block plus D, M, S, S, N makes 6 units, with S repeated: 6!/2! = 360. Inside the block, the letters A, I, I, O can arrange in 4!/2! = 12 ways.
Total = 360 × 12 = 4,320.
Example 3: Gap method
Five men and four women sit in a row. No two women may sit together. How many arrangements work?
Solution: Arrange the men first: 5! = 120. They create 6 gaps. Choose and arrange 4 women in those gaps: ⁶P₄ = 360.
Total = 120 × 360 = 43,200.
Example 4: Stars and bars
You give 12 identical chocolates to 4 children.
(a) Every child must get at least one. (b) A child may get none.
Solution:
(a) Give each child one chocolate, then distribute the remaining 8: ¹¹C₃ = 165.
(b) ¹⁵C₃ = 455.
Example 5: Complement method
A committee of 4 comes from 6 men and 5 women. It must include at least one woman. How many committees work?
Solution: Total committees: ¹¹C₄ = 330. Committees with no woman: ⁶C₄ = 15.
Answer = 330 − 15 = 315.
Example 6: Circular arrangement
Six people sit around a round table. Two particular people, X and Y, must sit together. In how many ways can they sit? How many ways if they must not sit together?
Solution: Treat X and Y as one unit. Five units sit around a table in (5−1)! = 24 ways. X and Y swap places inside the unit: 24 × 2 = 48.
All arrangements: 5! = 120. So not together: 120 − 48 = 72.
Example 7: Counting rectangles
How many rectangles, including squares, appear on an 8 × 8 chessboard?
Solution: A rectangle needs 2 of the 9 horizontal lines and 2 of the 9 vertical lines. That gives ⁹C₂ × ⁹C₂ = 36 × 36 = 1,296. Of these, 204 are squares.
CAT Permutation and Combination Practice Questions with Answers
Try each question for about two minutes before you check the answer table.
- How many 3-digit numbers can you form from the digits 0, 1, 2, 3, 4, 5, without repeating any digit, that are divisible by 5?
- In how many ways can you arrange the letters of the word COMMITTEE?
- A team of 5 comes from 6 boys and 4 girls. It must include at least 2 girls. How many teams are possible?
- Seven people sit around a circular table. Two particular people must not sit next to each other. In how many ways can they sit?
- Find the number of positive integer solutions of a + b + c + d = 15.
- Four boys and three girls sit in a row. All three girls must sit together. How many arrangements are possible?
- Twelve points lie in a plane. Five of them are collinear, and no other three are collinear. How many triangles can you form using these points as vertices?
- A person walks from (0, 0) to (6, 4), moving one unit right or one unit up at each step. How many paths pass through the point (3, 2)?
- In how many ways can you choose 3 different numbers from 1 to 20 so that their sum is even?
- In how many ways can you give 5 different gifts to 3 children so that every child gets at least one gift?
Answer key
| Q | Answer | Method |
|---|---|---|
| 1 | 36 | Last digit 0: 5 × 4 = 20. Last digit 5: 4 choices for the first digit and 4 for the middle = 16 |
| 2 | 45,360 | 9!/(2! 2! 2!), since M, T and E each repeat twice |
| 3 | 186 | ¹⁰C₅ = 252, minus 0 girls (6) and 1 girl (4 × ⁶C₄ = 60) |
| 4 | 480 | Total 6! = 720. Together 5! × 2 = 240. So 720 − 240 |
| 5 | 364 | Each variable at least 1: ¹⁴C₃ |
| 6 | 720 | Girls’ block plus 4 boys = 5 units: 5! × 3! |
| 7 | 210 | ¹²C₃ − ⁵C₃ = 220 − 10 |
| 8 | 100 | (0,0) to (3,2): ⁵C₂ = 10. (3,2) to (6,4): ⁵C₂ = 10. Product = 100 |
| 9 | 570 | All even: ¹⁰C₃ = 120. One even and two odd: 10 × ⁱ⁰C₂ = 450 |
| 10 | 150 | 3⁵ − 3 × 2⁵ + 3 = 243 − 96 + 3 |
After these, test yourself on real exam questions in the CAT previous year papers.
Common Mistakes in CAT Permutation and Combination
| Mistake | Fix |
|---|---|
| Treating identical items as different | Divide by the factorial of each repeated item’s count |
| Letting a number start with 0 | Fill the first place first, or subtract the cases with a leading zero |
| Using nPr for a selection | Run the order test before choosing a formula |
| Counting “at least” cases one by one | Use total minus the unwanted case |
| Forgetting the order inside a glued block | Multiply by the internal arrangements of the block |
| Using n! for seating around a table | Use (n−1)!, because rotations look identical |
| Miscounting gaps | n items in a row create n + 1 gaps |
How to Prepare P&C Before CAT 2026
With the exam on 29 November, you can still cover this topic in about two weeks. Keep revising Arithmetic and Algebra alongside it, as both are important areas in CAT Quantitative Aptitude. You can also strengthen your Algebra preparation with our guide on quadratic equations for CAT.
| Days | Focus |
|---|---|
| 1–3 | Factorials, the counting principle, nPr, nCr, the order test |
| 4–6 | Restrictions: glue, gap and complement methods, number formation |
| 7–9 | Circular arrangements, distributions, stars and bars, geometry counting |
| 10–12 | Mixed sets under a timer |
| 13–14 | Previous year questions and a full mock |
Use the CATMOCK formula book for quick revision and daily practice to keep the habit going. For timing your mocks, read when to start CAT mock tests.
Summary
- Permutation counts arrangements where order matters. Combination counts selections where order does not.
- Run the order test before you pick a formula.
- Three methods solve most restricted questions: glue for “together”, gap for “never together” and complement for “at least”.
- Modern Maths has a low and variable weightage in CAT, so a focused two-week plan is enough.
- Most errors come from double counting, identical items and leading zeros.
FAQs on CAT 2026 Permutation and Combination
1. Is permutation and combination important for CAT 2026?
It carries a modest weightage, because it sits in Modern Maths, which varies from year to year. It still deserves attention because it supports Probability and gives you quick marks when the question is clean.
2. How many questions come from permutation and combination in CAT?
No fixed number exists. Modern Maths contributes only a small share of the QA section, and the count of pure P&C questions changes each year. Check recent papers and exam analyses for the latest trend.
3. How do I decide between permutation and combination?
Ask whether swapping two chosen items changes the outcome. If it does, use a permutation. If it does not, use a combination.
4. What is the formula for circular permutation?
For n distinct people around a table, the count is (n−1)!. For a necklace or garland, where flipping gives the same arrangement, it is (n−1)!/2.
5. How many days do I need to prepare permutation and combination for CAT?
If your basics are in place, about 10 to 14 days of focused practice usually covers the essentials. Add more time if you are new to factorials and counting.
6. Can I skip permutation and combination in CAT?
You can skip questions in any topic, but dropping the whole of Modern Maths narrows your choices on exam day. A short, well-planned revision keeps that option open.
7. Where can I download the CAT permutation and combination formulas PDF?
Use the download link in this article for the formula sheet and practice set. The CatMock formula book also holds more quick-revision material.
Conclusion
CAT 2026 permutation and combination rewards clear thinking, not heavy calculation. Start with the order test, pick the matching method and check the condition before you count. Practise the seven examples and ten questions above, then move to previous year papers and timed mocks on CATMOCK. Do that, and counting questions stop being a risk and start looking like quick marks.
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