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  "body": "\n## Summary\n\nDouble counting is a proof technique in combinatorics.\nTo prove that two expressions are equal, one shows that both count the elements of the same finite set.\nEach expression comes from a different way to count that set.\n\nA common form uses a relation $R \\subseteq A \\times B$ between finite sets $A$ and $B$.\nFor $a \\in A$, let $d(a)$ be the number of $b \\in B$ with $(a, b) \\in R$.\nFor $b \\in B$, let $e(b)$ be the number of $a \\in A$ with $(a, b) \\in R$.\nThen both sums count the pairs in $R$, so\n$$\\sum_{a \\in A} d(a) = |R| = \\sum_{b \\in B} e(b).$$\n\n## Remarks\n\nThe [[handshaking lemma|0XAP8WHNES]] is double counting applied to the pairs of a vertex and an edge that contains it.\nThe [[symmetry of binomial coefficients|0R7JS3DP3W]] has a short proof by this technique.\n\n## Explanations\n\n### L1 human\n\nCount the same thing in two ways, and the two answers must be equal.\nFor example, in a class each pair of friends shakes hands once. If you add the number of handshakes of each person, you count each handshake twice, so the sum is even.\n",
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