11TH•Mathematics•Permutations And Combinations•7.4 Combinations

7.4 Combinations

Combinations, complementary combinations, and real-life applications.

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A combination of $r$ objects from $n$ is a subset of size $r$ (order does not matter).

{}^n C_r = C(n,r) = \binom{n}{r} = \dfrac{n!}{r!(n-r)!}

Proof. Each combination of $r$ objects yields $r!$ permutations, so ${}^n C_r \cdot r! = {}^n P_r$ .

Corollaries: ${}^n C_n = 1$ , ${}^n C_0 = 1$ .

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