Permutations and Combinations (P&C) is a fundamental JEE Maths chapter. In JEE Main 2024, 1-2 questions appeared from this chapter; JEE Advanced combines P&C with probability. This chapter also forms the foundation for Binomial Theorem.
Fundamental Counting Principle
- AND (both tasks): m x n ways (multiplication)
- OR (either task): m + n ways (addition)
Permutations nPr = n! / (n-r)!
Arrangements where ORDER matters.
| Case | Formula |
|---|---|
| All n objects | n! |
| Circular arrangement | (n-1)! |
| Necklace/Bracelet | (n-1)!/2 |
| With repetition allowed | n^r |
| n objects: p alike, q alike, r alike | n! / (p! x q! x r!) |
Combinations nCr = n! / [r!(n-r)!]
Selection where order does NOT matter.
Key Properties:
- nCr = nC(n-r)
- nCr + nC(r-1) = (n+1)Cr [Pascal Identity]
- Sum of all nCr = 2^n (total subsets)
- nPr = nCr x r!
Key Formulas
| Problem Type | Formula |
|---|---|
| Diagonals in n-sided polygon | n(n-3)/2 |
| Handshakes among n people | nC2 = n(n-1)/2 |
| Triangles from n non-collinear points | nC3 |
| Divisors of N = p^a x q^b x r^c | (a+1)(b+1)(c+1) |
Distribution of Objects
| Type | Formula |
|---|---|
| n distinct objects into r distinct boxes | r^n |
| n identical into r distinct boxes (empty allowed) | (n+r-1)C(r-1) |
| n identical into r distinct boxes (none empty) | (n-1)C(r-1) |
Restrictions
- Always together: Treat as 1 unit – (n-k+1)! x k!
- Never together: Total – (always together)
- Boys and girls alternate (row): 2 x n! x n!
Practice MCQs – Permutations and Combinations
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Last updated: April 2026 | JEE Gurukul – Concept-first approach to JEE Mathematics