Vandermonde's identity (${\sum }_{i=0}^k\left(\genfrac{}{}{0ex}{}{m}{i}\right)\left(\genfrac{}{}{0ex}{}{n}{k-i}\right)=\left(\genfrac{}{}{0ex}{}{m+n}{k}\right)$)

Choose a $k$-element subset of a disjoint union of sets of sizes $m$ and $n$. Partition the choices by the number $i$ of selected elements from the first set. The resulting count is the displayed convolution of binomial coefficients, where coefficients outside their admissible ranges are zero. This also proves the formula when $k>m+n$.