Counting Table

Cells $n$	distinct	distinct	identical	identical
Balls $k$	distinct	identical	distinct	identical
$0\le x_i\le 1$, $n\ge k$	$\frac{n!}{(n-k)!}=(n{)}_k$	$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$	1	1
$x_i\ge 0$	$n^k$	$\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)$		$p(k)$ ^[Partition]
$x_i\ge 1$	$n!\{\genfrac{}{}{0ex}{}{k}{n}\}$	$\left(\genfrac{}{}{0ex}{}{k-1}{n-1}\right)$		$p_n(k)$ ^[ [[Partition#Partition into k]]. Note, there, n is balls, and k is cells ]