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  "body": "\n## Proof\n\nWe argue by induction on $m \\ge 1$. The claim for $m$ is that the formula holds for this $m$, all $x_1, \\dots, x_m \\in R$ and all $n$. For an integer $c$ and $r \\in R$, $c \\cdot r$ is the repeated sum.\n\n1. Let $m = 1$. The only tuple $(k_1)$ with $k_1 = n$ is $(n)$, and the [[multinomial coefficient|00K0RF9BFY]] is $\\frac{n!}{n!} = 1$. So the right side is $x_1^n$, which is the left side.\n2. Suppose the claim holds for $m$. Let $x_1, \\dots, x_{m+1} \\in R$, and put $s = x_1 + \\dots + x_m$.\n3. The ring $R$ is [[commutative|0GYGHKX834]]. By the [[binomial theorem|0EHWPWGHB2]], $(s + x_{m+1})^n = \\sum_{j=0}^{n} \\binom{n}{j} s^j x_{m+1}^{n-j}$.\n4. By the induction hypothesis with exponent $j$, $s^j = \\sum_{k_1 + \\dots + k_m = j} \\binom{j}{k_1, \\dots, k_m} x_1^{k_1} \\cdots x_m^{k_m}$.\n5. Substituting and distributing gives $(s + x_{m+1})^n = \\sum_{j=0}^{n} \\sum_{k_1 + \\dots + k_m = j} c \\cdot x_1^{k_1} \\cdots x_m^{k_m} x_{m+1}^{n-j}$, with $c = \\binom{n}{j} \\binom{j}{k_1, \\dots, k_m}$.\n6. By the [[factorial formula|033KTKKRV6]], $\\binom{n}{j} = \\frac{n!}{j!\\,(n-j)!}$. The definition gives $\\binom{j}{k_1, \\dots, k_m} = \\frac{j!}{k_1! \\cdots k_m!}$.\n7. The factor $j!$ cancels, so $c = \\frac{n!}{k_1! \\cdots k_m!\\,(n-j)!} = \\binom{n}{k_1, \\dots, k_m, n-j}$.\n8. Put $k_{m+1} = n - j$. The pairs $(j, (k_1, \\dots, k_m))$ with $0 \\le j \\le n$ and $k_1 + \\dots + k_m = j$ correspond to the $(m+1)$-tuples with $k_1 + \\dots + k_{m+1} = n$.\n9. The inverse of this correspondence sends $(k_1, \\dots, k_{m+1})$ to $j = k_1 + \\dots + k_m = n - k_{m+1}$. So each $(m+1)$-tuple appears exactly once.\n10. Rewriting the double sum of step 5 with steps 7 to 9 gives $(x_1 + \\dots + x_{m+1})^n = \\sum_{k_1 + \\dots + k_{m+1} = n} \\binom{n}{k_1, \\dots, k_{m+1}} x_1^{k_1} \\cdots x_{m+1}^{k_{m+1}}$.\n11. This completes the induction.\n\n## Idea\n\nGroup the first $m$ terms into one term $s$ and expand $(s + x_{m+1})^n$ with the binomial theorem.\nEach power of $s$ is expanded by the induction hypothesis.\nThe binomial coefficient times the smaller multinomial coefficient is the larger multinomial coefficient, because the factor $j!$ cancels.\n",
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