Permutations and Combinations
Board: JEE Main | Exam: JEE Main
JEE Main study notes for Permutations and Combinations by Ajay Yadav. This chapter covers all concepts required for JEE Main with problem-solving techniques.
Key Concepts
Counting Principles
Multiplication principle, addition principle. Factorial: n!. nPr = n!/(n-r)!. nCr = nPr/r!.
Combinations
nCr = nC(n-r). nC0 = nCn = 1. Pascal’s identity: nCr + nC(r-1) = (n+1)Cr.
Distributions
Number of ways to distribute n distinct objects into r distinct boxes: rⁿ. Identical objects into distinct boxes: C(n+r-1, r-1).
Important Results
Number of diagonals = nC2 – n. Number of rectangles in m×n grid = mc2 × nc2. Derangements: !n = n!Σ(-1)k/k!
Important Formulas
| nPr | n!/(n-r)! |
| nCr | n!/[r!(n-r)!] |
| nCr = nC(n-r) | True |
| Pascal | nCr + nC(r-1) = (n+1)Cr |
Solved Examples
Example 1: How many 3-digit numbers with distinct digits?
Solution: 9×9×8 = 648 (first digit ≠0).
Example 2: Find number of ways to select 3 from 10 items.
Solution: C(10,3) = 120.
Practice Questions
- How many words from “DELHI” starting with D?
- Find n if nC5 = nC7.
- In how many ways can 5 boys and 3 girls sit in a row if no two girls sit together?
- Find number of rectangles in 8×8 chessboard.
- Number of ways to distribute 10 identical coins to 4 children.