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  "body": "\n## Statement\n\nLet $n$ be a natural number.\nThen\n$$\\sum_{k=0}^{n} \\binom{n}{k} = 2^n.$$\n\n## Remarks\n\nFor $n = 4$: $1 + 4 + 6 + 4 + 1 = 16$.\nThe identity is the [[binomial theorem|0EHWPWGHB2]] with $x = y = 1$.\nCombinatorially, both sides count the subsets of an $n$-element set. The [[number of subsets of a finite set|0C5RDJE73X]] is $2^n$, and $\\binom{n}{k}$ of them have size $k$.\n\n## Explanations\n\n### L1 human\n\nThe numbers in row $n$ of [[Pascal's triangle|05DME4TZ3K]] add up to $2^n$.\nThis is because each subset of an $n$-element set has some size $k$ from $0$ to $n$, and there are $2^n$ subsets in total.\n",
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