Combinations and arrangements

Given $n$ distinct objects, the number of ways to arrange $r$ of them in order (arrangements or permutations) is $P(n, r) = n imes (n-1) imes ext{...} imes (n-r+1) = \frac{n!}{(n-r)!}.$ The number of ways to choose $r$ objects from $n$ without caring about order (combinations) is $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}.$

Important properties

• Arrangements (permutations) count ordered selections.

• Combinations count unordered selections.

• For each combination, there are $r!$ arrangements.

• The sum of all combinations for $r = 0$ to $n$ is $2^n$.