Nested-subset binomial identity ($\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{k}{\ell }\right)=\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)\left(\genfrac{}{}{0ex}{}{n-\ell }{k-\ell }\right)$)

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.