Permutations

The number of different arrangements of a k-element subset of the n-element set S. $$P(n,k) = n \times (n - 1) \times \dots \times (n-k + 1) = \frac{n!}{(n-k)!}$$

Cobinations

The number of unique, ordered arrangements of a k-element subset of the n-element set S. Also referred to as Binomial Coefficients $$C(n,k) = {n \choose k} = \frac{n!}{k!(n-k)!}$$

Pascal's Triangle

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Binomial Expansion

$$(x + y)^{n} = \sum_{k = 0}^{n} {} {n \choose k}x^{n-k}y^{k}$$

Permutations of multisets

The number of arrangements of the n-element multiset,M in which each element appears exactly as often as the multiplicity. $${n \choose k_1,k_2, \dots ,k_i} = \frac{n}{k_1!k_2! \dots k_i!}$$