Permutation Combination Formulas for CAT 2026 + 20 PYQs
Permutation and combination formulas for CAT 2026 cover one of the most confused topics in QA, where the same question can be solved with a permutation or a combination depending on a single line of wording. The trap is rarely formula recall. The trap is mis-recognising whether the question rewards order (permutation) or selection (combination). This cheatsheet pins 20 formulas across five blocks and ends with 20 PYQ-style questions that mirror how CAT setters actually use the topic.
CAT typically includes one to two direct PnC questions per paper, often linked to a probability question that uses the same setup. Mastering PnC therefore also lifts probability scores. The cheatsheet below is organised by the kind of question pattern each formula solves, not by alphabetical formula order, so each block doubles as a recognition guide.
The Arrangement vs Selection Test: How to Read a PnC Question
Before any formula gets touched, every PnC question must pass a single test: does order matter in the final answer? If reordering the chosen items produces a different outcome, the question is a permutation. If reordering produces the same outcome, the question is a combination. This one-sentence test resolves about ninety percent of the recognition gap CAT aspirants have on PnC questions.
Words that usually signal permutation: arrange, sequence, rank, position, line up, sit in a row, form a code. Words that usually signal combination: choose, select, pick, group, committee, team, set, draw. The trap CAT setters use is wording a permutation question with combination language, or vice versa, expecting candidates to skip the order test. The 20 PYQ-style questions below walk through several of these traps.
The 20 Permutation and Combination Formulas for CAT 2026
The cheatsheet groups 20 formulas into five blocks. Each block carries a recognition cue: a sentence describing the kind of question that triggers it. Working block-by-block embeds the recognition habit that the arrangement vs selection test depends on.
Block 1 — Counting Basics (3 formulas)
Counting basics are the foundation under every PnC question. Even advanced restrictions and distribution problems ultimately invoke one of these three rules, which makes them the first block to memorise. Recognition cue: any question that asks how many ways something can be done in stages.
| # | Formula | Recognition cue |
|---|---|---|
| 1 | n! = n × (n − 1) × ... × 2 × 1 | Factorial of n. |
| 2 | Multiplication principle: if A in m ways and B in n ways, then A then B in m × n ways | Sequential choices. |
| 3 | Addition principle: if A in m ways or B in n ways (mutually exclusive), then m + n ways | Either-or choices. |
Block 2 — Permutations (4 formulas)
Permutation formulas apply when order matters. Whether items are distinct, repeated, or arranged in a circle, one of the four formulas below covers it. Recognition cue: the question is about arrangement, sequence, position, or ranking of selected items.
| # | Formula | Use case |
|---|---|---|
| 4 | nPr = n! / (n − r)! | Arrange r items from n distinct items. |
| 5 | n! / (p1! × p2! × ... × pk!) | Permutations with repeated items. |
| 6 | (n − 1)! | Circular permutations of n distinct items. |
| 7 | (n − 1)! / 2 | Circular with reflection (beads on a necklace). |
Block 3 — Combinations (4 formulas)
Combination formulas apply when order does not matter. The four identities below cover every selection question, including the binomial summation that links PnC to binomial theorem. Recognition cue: the question is about choosing, selecting, or forming a group.
| # | Formula | Use case |
|---|---|---|
| 8 | nCr = n! / (r! × (n − r)!) | Select r items from n distinct items. |
| 9 | nCr = nC(n − r) | Symmetry identity for selection. |
| 10 | nC0 + nC1 + ... + nCn = 2n | Sum of all subsets. |
| 11 | nCr + nC(r − 1) = (n + 1)Cr | Pascal identity for adjacent combinations. |
Block 4 — Restrictions and Conditions (5 formulas)
Restrictions are where CAT setters get creative. The five techniques below cover most restricted PnC questions: at least, at most, sit together, not adjacent, and never-includes constraints. Recognition cue: the question imposes a rule on the arrangement.
| # | Formula | Use case |
|---|---|---|
| 12 | At least k: total − (at most k − 1) | Subtraction trick for at-least. |
| 13 | Group method: treat "sit together" group as one unit, then multiply by internal arrangements | Items must be adjacent. |
| 14 | Gap method: arrange other items first, then place restricted items in gaps | Items must not be adjacent. |
| 15 | Specific item included: count by including the item, then arrange the rest | Item must be in. |
| 16 | Specific item excluded: arrange without that item entirely | Item must be out. |
Block 5 — Distribution and Advanced (4 formulas)
Distribution formulas distribute identical items into groups or handle inclusion-exclusion overlaps. The four below cover the highest-leverage advanced techniques on CAT PnC. Recognition cue: the question distributes identical items or has overlapping conditions.
| # | Formula | Use case |
|---|---|---|
| 17 | Stars and bars: (n + r − 1) C (r − 1) | Distribute n identical items into r groups (zero allowed). |
| 18 | Stars and bars (each ≥ 1): (n − 1) C (r − 1) | Distribute with at-least-one rule. |
| 19 | Inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B| | Overlapping conditions. |
| 20 | Number of factors of N = (a + 1)(b + 1)(c + 1)... | Counting factors via prime factorisation. |
The Three Highest-Frequency Traps in CAT PnC Questions
Three traps recur in CAT PnC. The first is mis-tagging arrangement vs selection, covered above. The second is forgetting to subtract restrictive cases when the question asks "at least" or "at most." The fix is the subtraction principle: total minus the complement. The third is double-counting in restriction questions, where a candidate accidentally counts the same arrangement twice through overlapping cases.
The fix for double-counting is inclusion-exclusion: count each condition separately, then subtract overlaps. For most CAT questions, two-set inclusion-exclusion is enough. Three-set inclusion-exclusion appears occasionally but is rare. Drilling each trap with 3-5 questions builds the reflex to catch the error mid-solve.
20 Must-Solve CAT PnC Questions
These 20 questions cover all five blocks. Each is tagged with the block and the formula it tests. The drill: solve under timed conditions, target under 90 seconds per question, and consult the cheatsheet only after attempting.
A menu has 4 starters, 5 main courses, and 3 desserts. How many meal combinations?
4 × 5 × 3 = 60. Answer: 60
A box has 7 red and 5 blue balls. In how many ways can one ball be selected?
7 + 5 = 12. Answer: 12
How many ways to arrange 4 books out of 10 on a shelf?
10P4 = 10!/(6!) = 10 × 9 × 8 × 7 = 5040. Answer: 5040
How many distinct arrangements of letters in MISSISSIPPI?
11!/(4! × 4! × 2!) = 39916800/(24 × 24 × 2) = 34650. Answer: 34650
In how many ways can 7 people sit around a round table?
(7 − 1)! = 6! = 720. Answer: 720
In how many ways can 5 students be selected from 12?
12C5 = 792. Answer: 792
How many subsets does a set of 8 elements have?
28 = 256. Answer: 256
If nC4 = nC9, find n.
By symmetry nCr = nC(n-r), so 4 + 9 = n. n = 13. Answer: 13
In how many ways can 6 people sit in a row if 2 specific people must sit together?
Treat the pair as one unit: 5! arrangements × 2! internal = 120 × 2 = 240. Answer: 240
In how many ways can 4 boys and 3 girls stand in a row so that no two girls are adjacent?
Arrange 4 boys: 4! = 24. There are 5 gaps for girls: 5P3 = 60. Total = 24 × 60 = 1440. Answer: 1440
From 10 students, how many ways to form a team of at least 8?
10C8 + 10C9 + 10C10 = 45 + 10 + 1 = 56. Answer: 56
From 12 players, choose 5 such that captain X is always included.
Fix X, choose 4 more from remaining 11: 11C4 = 330. Answer: 330
From 12 players, choose 5 such that player Y is never included.
Choose 5 from remaining 11: 11C5 = 462. Answer: 462
Distribute 10 identical chocolates among 4 children (each can get zero or more).
(10 + 4 − 1) C (4 − 1) = 13C3 = 286. Answer: 286
Distribute 10 identical chocolates among 4 children with each getting at least 1.
Give 1 each first (4 used), distribute remaining 6: (6 + 4 − 1) C (4 − 1) = 9C3 = 84. Answer: 84
How many positive factors does 360 have?
360 = 23 × 32 × 5. Factors = (3+1)(2+1)(1+1) = 24. Answer: 24
In a class of 50, 30 like maths, 25 like physics, 15 like both. How many like neither?
Like at least one = 30 + 25 − 15 = 40. Like neither = 50 − 40 = 10. Answer: 10
How many 4-digit numbers can be formed with digits 1-9 (no repetition) such that the number is even?
Last digit even: 2, 4, 6, or 8, so 4 choices. Remaining 3 places from 8 digits: 8P3 = 336. Total = 4 × 336 = 1344. Answer: 1344
If 3 cards are drawn from a 52-card deck, find P(all three are hearts).
P = 13C3 / 52C3 = 286 / 22100 = 11/850. Answer: 11/850
A committee of 5 is to be formed from 6 men and 5 women such that at least 2 women are included. How many ways?
Total = (2W3M) + (3W2M) + (4W1M) + (5W0M) = 5C2×6C3 + 5C3×6C2 + 5C4×6C1 + 5C5×6C0 = 200 + 150 + 30 + 1 = 381. Answer: 381
Slot PnC Into a CAT 2026 Plan That Matches Your Level
Permutation and combination is one of 22 Quants topics that need to fit into a CAT 2026 plan. A diagnostic-driven sequence puts PnC where it earns the most marks per study hour, given your starting level.
Build My PnC + Probability PlanWhere PnC Fits in the Larger CAT 2026 Quants Sequence
PnC is a self-contained topic that does not depend on most other Quants chapters. This makes it a flexible block in any plan. A sensible placement is month two of a 9-month plan, alongside Probability, so the two linked topics can be drilled together. The 20 PnC formulas plus the 12 to 15 Probability formulas form a single mental cluster, and CAT setters often pair them in linked question sets. Sequencing them together produces stronger pattern recognition than studying each separately.
For a working professional or college student, PnC suits short study blocks because each formula maps to a clean question type. Each block can cover 4-5 formulas plus 3 worked questions in a 45-minute session. For sequencing the rest of the QA syllabus around PnC, our CAT exam guide walks through topic priorities, and the CAT 2026 waitlist details page explains how the personalised planner builds your topic sequence.
Three Habits That Make PnC Feel Routine on CAT 2026
Once 20 formulas are memorised, three habits separate aspirants who solve PnC in 60 seconds from those who take three minutes. Habit one: order test first. Write "Order: Y/N" on the rough sheet before any arithmetic. Habit two: condition tagging. If the question says "at least," "at most," "must include," or "must exclude," circle the constraint before applying the formula. Habit three: complement check. For "at least" questions with more than 2-3 cases, compute the complement (total minus excluded) instead of summing cases directly. These three install through timed drill. The CAT practice questions library on Optima Learn tags PnC problems by block so each habit can be drilled separately.
Common Doubts About CAT PnC Preparation
How many PnC questions appear in CAT 2026?
Expect 1 to 2 direct PnC questions and 1 linked Probability question per CAT QA section. That makes the PnC + Probability cluster a 3 to 4 mark contributor on average. Given that 20 formulas cover the topic, this is a strong return on the focused week of preparation, especially if PnC is paired with Probability in the study plan.
Is the stars and bars method worth memorising?
Yes. Stars and bars converts distribution questions, which look complex on paper, into a single-step combination calculation. CAT uses stars and bars about once every two papers, and recognising the pattern gives a 60-second solve. The two variants, one with zero-or-more and one with at-least-one, cover virtually every distribution question CAT setters use.
Should I memorise the Pascal identity?
The Pascal identity nCr plus nC(r-1) equals (n+1)Cr is useful but not essential. It appears occasionally in CAT through algebraic simplification of combination sums. Memorising it costs nothing and helps in maybe one paper out of three. Worth the small effort for the marginal upside.
How do I revise PnC one week before CAT 2026?
A one-week revision plan: day one, re-read the 20 formulas and tag each with its block. Day two to four, attempt one block per day with five timed questions. Day five, run 20 mixed questions under exam conditions. Day six, review every error. Day seven, scan the cheatsheet for 20 minutes only before the exam.
Final note. Permutation and combination formulas for CAT 2026 reduce to 20 formulas across 5 blocks, with the arrangement vs selection test resolving most of the recognition gap. Drill block-by-block, pair with Probability, and the CAT score predictor alongside mocks will track the improvement.
