Vandermonde's Identity states that for non-negative integers $n$, $m$, and $r$, the sum of the products of binomial coefficients is given by $$\sum_{k=0}^{r} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r}$$. This identity provides a way to relate two separate groups' combinations to a combined group, showcasing a deep connection between combinatorial counting and binomial coefficients.
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