CAT Quant · Study & Practice

Permutation & Combination

AreaModern Maths DifficultyModerate–Hard CAT weightage2–4 questions (often paired with Probability; high-yield in XAT/SNAP)

Permutation and Combination is the gateway to all of counting — and to Probability, which sits right next to it on every CAT and XAT paper. The entire chapter rests on one decision: does order matter? If rearranging the chosen items creates a genuinely new outcome (seats in a row, a 3-digit code, a podium of gold-silver-bronze), you are counting arrangements — permutations. If only the group matters and shuffling it changes nothing (a committee, a hand of cards, a salad of fruits), you are counting selections — combinations. Once that question is settled, two formulas do almost everything: nPr = n!/(n−r)! for arranging r of n, and nCr = n!/(r!(n−r)!) for choosing r of n. CAT rarely asks a clean "how many ways"; it buries the count inside restrictions — items that must sit together, items that must stay apart, repeated letters, people around a round table. This chapter builds the full toolkit: the fundamental counting principle, arrangements with repetition, circular and necklace arrangements, and the gap and grouping methods that crack restricted problems fast.

Topics

1 Arrangements 4 worked · 5 practice · 8-Q test Start → 2 Selections 4 worked · 5 practice · 8-Q test Start → 3 Circular Arrangements 4 worked · 5 practice · 8-Q test Start → 4 Restricted Arrangements 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • First decide: does order matter? Order ⇒ permutation (nPr); group only ⇒ combination (nCr). This one question solves half the chapter.
  • Use nCr = nC(n−r) to compute the easier side: 10C8 is just 10C2 = 45.
  • Draw blanks for digit/code problems and write valid choices per slot, then multiply — watch the leading-zero and repetition rules.
  • Round table = (n−1)!; necklace/garland (can flip) = (n−1)!/2; numbered seats or a fixed head = n! (no free rotation).
  • Together ⇒ glue into a block: (units)! × (internal)!. Apart ⇒ gap method: arrange the rest, then drop items into the gaps.
  • Repeated letters ⇒ divide n! by the factorial of each repeat count (BANANA = 6!/(3!2!) = 60).

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Using a permutation when order does not matter (counting a committee as if seats were ordered) — inflates the answer by r!.
  • Forgetting the leading digit cannot be 0 in number-formation problems.
  • Using n! for a round table instead of (n−1)!, or forgetting the extra ÷2 for a necklace/garland.
  • Not dividing by the factorials of repeated letters (treating BANANA’s three A’s as distinct).
  • Adding cases joined by "and" or multiplying cases joined by "or" — it is the reverse: "and" multiplies, "or" adds.

📈 CAT exam insight & PYQ analysis

CAT keeps P&C lean but tricky — usually 1–2 questions, frequently fused with Probability so the counting is only half the battle. The recurring favourites are arrangements of a word with repeated letters, restricted seatings (together / never together via gap and grouping), formation of numbers under divisibility or even/odd conditions, and selection problems with "at least" phrasing solved by the complement. XAT and SNAP lean harder on circular arrangements and casework. Difficulty is Moderate–Hard: the formulas are short, but the marks go to students who classify the problem correctly and pick the gap, grouping, or complement method without fumbling.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

nPr formula?Tap to reveal
n! / (n − r)!
nCr formula?Tap to reveal
n! / (r!(n − r)!)
Relation between nPr and nCr?Tap to reveal
nPr = nCr × r!
Arrangements of n items with repeats p, q, …?Tap to reveal
n! / (p! q! …)
Circular arrangement of n distinct people?Tap to reveal
(n − 1)!
Necklace / garland of n distinct beads?Tap to reveal
(n − 1)! / 2
nCr = nC?Tap to reveal
nC(n − r)
Total subsets of an n-element set?Tap to reveal
2ⁿ
"At least one" selection from n items?Tap to reveal
2ⁿ − 1
Distinct arrangements of BANANA?Tap to reveal
6!/(3!2!) = 60
Method for "items must be together"?Tap to reveal
Glue into a block: (units)! × (internal)!
Method for "no two items together"?Tap to reveal
Gap method: arrange the rest, place items in the gaps

📌 Quick revision

Counting starts with one question: does order matter? Order means permutations, nPr = n!/(n−r)!; pure grouping means combinations, nCr = n!/(r!(n−r)!), and remember nCr = nC(n−r). The fundamental counting principle multiplies independent steps. Repeated items divide n! by the factorial of each repeat. A round table is (n−1)!, a necklace or garland (n−1)!/2, but numbered or headed seats revert to n!. For restrictions, glue items that must stay together and use the gap method for items that must stay apart, or count the complement for "at least" cases. Classify first, then the formula is short.

Chapter test

📝 Permutation & Combination — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Permutation & Combination when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards