Permutations & Combinations

Concept overview

Counting problems break into two categories: permutations (order matters) and combinations (order doesn't). The number of permutations of n distinct objects is n!. Choosing r of them in order gives P(n, r) = n!/(n − r)!. Choosing r without regard to order gives C(n, r) = n!/(r!(n − r)!).

The fundamental counting principle multiplies choices at successive stages: if stage 1 has m options and stage 2 has n options, the total is m × n. This is the base for almost every counting problem.

Circular permutations of n distinct objects give (n − 1)! arrangements — we fix one object to break the rotational symmetry. If reflections are also equivalent (necklace-type problems), divide by 2 further.

Permutations with restrictions (some objects together, some apart, fixed positions) use the technique of treating grouped objects as single units and then multiplying by internal arrangements. Circular arrangements with constraints multiply the general (n − 1)!-count by the fraction of arrangements satisfying the constraint.

Combinations with repetition — choosing r items from n types with repetition allowed — give C(n + r − 1, r). This is used for the classic 'ball-in-box' problems.

Key formulas & concepts

• Permutations: P(n, r) = n!/(n − r)!
• Combinations: C(n, r) = n!/(r!(n − r)!)
• Circular permutations: (n − 1)! for n distinct objects
• Arrangements with identical items: n! / (n₁! n₂! ... nₖ!)
• Selections with repetition: C(n + r − 1, r)
• Pigeonhole principle: n + 1 objects in n boxes ⇒ at least one box has ≥ 2 objects

Common mistakes to avoid

• Confusing permutations with combinations — always ask whether order matters.
• Forgetting to divide by internal-permutation factorials when items repeat.
• Applying n! to circular arrangements — use (n − 1)! for circles.
• Ignoring symmetry (necklace / bracelet problems require dividing further by 2).

JEE Main relevance

JEE Main asks 1 counting problem every paper — usually a straightforward arrangement or selection with a specific constraint.

JEE Advanced relevance

Advanced problems introduce partitions or use inclusion-exclusion for arrangements with multiple forbidden patterns.

Previous-year questions (PYQ)

PYQs cluster around 'arrangement of letters' and 'selection with restriction' archetypes. Practise both.