06Class 11 Maths

Permutations and Combinations formulas

Counting formulas for Permutations and Combinations. The hardest part is deciding whether order matters, and the formulas are the easy bit.

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All Maths formulas

Core formulas

Factorial: $\displaystyle n! = n(n - 1)(n - 2)\cdots 1, \quad 0! = 1$
Permutations: $\displaystyle {}^nP_r = \frac{n!}{(n - r)!}$
When order matters.
Combinations: $\displaystyle {}^nC_r = \frac{n!}{r!(n - r)!}$
When order doesn't matter.
Link between them: $\displaystyle {}^nP_r = r!\;{}^nC_r$

Useful results

Complement: $\displaystyle {}^nC_r = {}^nC_{n - r}$
Pascal's rule: $\displaystyle {}^nC_r + {}^nC_{r - 1} = {}^{n + 1}C_r$
Arrangements with repeats: $\displaystyle \frac{n!}{p_1!\,p_2!\cdots p_k!}$
p₁ things alike, p₂ alike, and so on.
With repetition allowed: $\displaystyle n^r$
Arranging r places from n things.

Where students slip up

Using P when order doesn't matter. Choosing a team is a combination, and ranking it is a permutation.
Adding when you should multiply. Use multiplication for 'and', and addition for 'or'.

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