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  "body": "\n## Statement\n\nLet $R$ be a commutative ring, let $x_1, \\dots, x_m \\in R$, and let $n$ be a natural number.\nThen\n$$(x_1 + \\dots + x_m)^n = \\sum_{k_1 + \\dots + k_m = n} \\binom{n}{k_1, \\dots, k_m} x_1^{k_1} \\cdots x_m^{k_m}.$$\nThe sum runs over all $m$-tuples $(k_1, \\dots, k_m)$ of natural numbers with sum $n$, and the coefficients are [[multinomial coefficients|00K0RF9BFY]].\n\n## Remarks\n\nFor $m = 2$ it is the [[binomial theorem|0EHWPWGHB2]].\nExample: $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$.\nWith all $x_i = 1$ it gives $\\sum \\binom{n}{k_1, \\dots, k_m} = m^n$.\n\n## Explanations\n\n### L1 human\n\nTo expand a power of a sum of $m$ terms, pick one term from each of the $n$ factors.\nThe monomial $x_1^{k_1} \\cdots x_m^{k_m}$ appears once for each way to decide which factors give which term.\n",
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