Vandermonde Identity

The Vandermonde identity (or Vandermonde convolution): $$\binom{m+n}{r} = \sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k}$$ Intuition: Choose $r$ items from a pool of $m$ red and $n$ blue objects. You can take $k$ red and $r-k$ blue for each valid $k$. When to use: Problems where you're choosing from two distinct groups, or when you see a sum of products of binomial coefficients that you need to simplify. Special case: $m = n = r$ gives $\binom{2n}{n} = \sum_{k=0}^{n} \binom{n}{k}^2$. Alternative approach: Double counting or a combinatorial argument (the "committee from two clubs" story) derives this from first principles without memorizing the formula.