 Nested-subset binomial identity

For $0\le \ell \le k\le n$, count pairs of subsets $L\subseteq K\subseteq X$ with $|X|=n$, $|L|=\ell$ and $|K|=k$. Choosing $K$ before $L$ gives the left product of binomial coefficients. Choosing $L$ and then the $k-\ell$ additional elements of $K$ gives the right product. This double counting proof explains the identity without division by factorials.