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  "body": "\n## Statement\n\nLet $m$, $n$ and $r$ be natural numbers.\nThen\n$$\\binom{m + n}{r} = \\sum_{k=0}^{r} \\binom{m}{k} \\binom{n}{r - k}.$$\n\n## Remarks\n\nCombinatorial proof: take a set of $m$ men and $n$ women. An $r$-element committee has some number $k$ of men and $r - k$ women.\nWith $m = n = r$, and the [[symmetry of binomial coefficients|0R7JS3DP3W]], it gives $\\binom{2n}{n} = \\sum_{k=0}^{n} \\binom{n}{k}^2$.\n\n## Explanations\n\n### L1 human\n\nTo choose $r$ people from two groups together, choose $k$ from the first group and $r - k$ from the second, for each possible $k$.\nThen add the counts for all values of $k$.\n",
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  "name": "Vandermonde's identity",
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