User:IssaRice/Summary of counting techniques

Let $A$ be a set with $n$ elements, and let $B$ be a set with $m$ elements.

Description	Set representing counting problem	number of ways to count
	$\{(a_1,\dots ,a_k):a_1,\dots ,a_k\in A\}$	$n^k$
	$\{\{a_1,\dots ,a_k\}:a_1,\dots ,a_k\in A\}$
	$\{(a_1,\dots ,a_k):a_1,\dots ,a_k\in A\text{ and all }{a}_i\text{ distinct}\}$	$P(n,k)=\frac{n!}{(n-k)!}=n(n-1)\cdots (n-(k+1))$
	$\{\{a_1,\dots ,a_k\}:a_1,\dots ,a_k\in A\text{ and all }{a}_i\text{ distinct}\}$	$(\genfrac{}{}{0ex}{}{n}{k})=P(n,k)/(k!)=\frac{n!}{k!(n-k)!}$
	$\{(a,b):a\in A\text{ and }b\in B\}$	$nm$