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  "body": "\n## Statement\n\nLet $n$ be a natural number.\nThen the [[complete graph|03WKB2H474]] $K_n$ has exactly\n$$\\binom{n}{2} = \\frac{n (n - 1)}{2}$$\nedges.\n\n## Remarks\n\nThe edges of $K_n$ are exactly the $2$-element subsets of the vertex set. The [[number of k-element subsets|0BFMR2FQ8B]] gives the count.\nThe [[handshaking lemma|0XAP8WHNES]] gives the same count: $n (n - 1) = 2 |E|$.\nEach simple graph on $n$ vertices has at most $\\binom{n}{2}$ edges.\n\n## Explanations\n\n### L1 human\n\nIf $n$ people all shake hands with each other, there are $\\frac{n(n-1)}{2}$ handshakes.\nEach person shakes $n - 1$ hands, and this counts each handshake twice.\n",
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