Permutations & Combinations: JEE Main Complete Guide

Permutations and Combinations is a chapter that tests logical thinking more than formula memorisation. JEE Main contributes two to three questions from it every session, and while the core formulae are simple, the application requires careful problem decomposition. The student who understands the multiplicative principle at a deep level can handle any P&C question; the one who memorises formulae without understanding the logic gets stuck on the first non-standard setup. This guide builds both the formulae and the logic.

The Fundamental Principle and nPr

If a task can be done in m ways and a second, independent task in n ways, both tasks together can be done in m×n ways (Multiplication Principle). If either task can be done (mutually exclusive), the count is m+n (Addition Principle). Every P&C problem reduces to applying these two principles correctly. A permutation is an ordered arrangement: nPr = n!/(n−r)! counts the number of ways to choose and arrange r objects from n distinct objects. The number of ways to arrange all n objects is n! (n factorial).

Arrangements with restrictions: if certain items must be together, treat them as a single super-item. If certain items must not be together, count total arrangements minus arrangements where they are together. These two moves — grouping and complementary counting — resolve the vast majority of arrangement problems. To check your speed on these foundational types, take a free P&C mock test.

Combinations: nCr and Its Properties

A combination is an unordered selection: nCr = n!/(r!(n−r)!) counts ways to choose r objects from n when order does not matter. Key identities: nCr = nC(n−r) (choosing r to include = choosing n−r to exclude), nCr + nC(r−1) = (n+1)Cr (Pascal's rule). These identities often allow simplification without full calculation. The number of ways to choose at least one item from n = 2ⁿ − 1 (all subsets minus the empty set). JEE uses "at least" problems frequently — complement counting (total minus restriction) is almost always faster than direct counting.

Division and distribution problems: dividing n distinct objects into groups of p, q, r (where p+q+r=n) gives n!/(p!q!r!) arrangements if the groups are labelled (distinguishable) or n!/(p!q!r!k!) if k groups have the same size (undistinguishable groups). These multinomial coefficients appear in both P&C and Probability chapters.

Circular Arrangements and Identical Objects

In a circular arrangement of n distinct objects, one object is fixed to remove the overcounting of rotations: (n−1)! arrangements. If the circle is a necklace (can be flipped), further divide by 2: (n−1)!/2. The concept of fixing one position is the key — without it, rotations of the same arrangement are counted as different, which is incorrect for circular problems.

Arrangements of objects where some are identical: with p identical items of one type, q of another, and the rest distinct, total arrangements = n!/(p!q!). This formula reduces the overcounting from treating identical items as distinct. JEE tests both the formula and the reasoning — why do we divide by p! and q!? Because p! arrangements of the identical items among themselves are indistinguishable. For the probability extension of these counting methods, see our probability guide. For the binomial theorem connections, see our binomial coefficients guide.