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Cheat sheet · JEE Preparation Mathematics

Permutations and Combinations cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 11 · Chapter 6Permutations and Combinations

Permutation = arrangement (order matters); Combination = selection (order doesn't).

1. Counting Principle

If one event can occur in m ways and another in n ways, both together can occur in m × n ways.

n!=n(n−1)⋯1n! = n(n - 1)\cdots10!=10! = 1

2. Formulas

 nPr=n!(n−r)!\,{}^nP_r = \dfrac{n!}{(n - r)!} nCr=n!r!(n−r)!\,{}^nC_r = \dfrac{n!}{r!(n - r)!}
 nCr=nCn−r\,{}^nC_r = {}^nC_{n - r} nCr+nCr−1=n+1Cr\,{}^nC_r + {}^nC_{r - 1} = {}^{n + 1}C_r
n!p1! p2!⋯\dfrac{n!}{p_1!\,p_2!\cdots} (some objects alike)

With repetition: nrn^r arrangements.

3. Worked Ideas

Letters of BANANA: 6!/(3! 2!) = 60.

Choose 3 of 7 people: ⁷C₃ = 35.

4. Key Points

  • ‘Arrange’ ⇒ P; ‘select’ ⇒ C.
  • Together: treat as one block.
  • ⁿPᵣ = r! ⁿCᵣ.
  • Read ‘at least’ carefully: add cases.

Remember: m × n rule | nPr vs nCr | identical objects

Permutations and Combinations: questions and notes

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