Kommutativer Halbring/Multinomialsatz/Fakt

Der Multinomialsatz (kommutativer Halbring)

Es sei ${}R$ ein kommutativer Halbring und seien ${}x_{1},\ldots ,x_{k}\in R$ Elemente und ${}n\in \mathbb {N}$ .

Dann ist

${}{\left(x_{1}+\cdots +x_{k}\right)}^{n}=\sum _{r_{1}+\cdots +r_{k}=n}{\binom {n}{r_{1},\dotsc ,r_{k}}}x_{1}^{r_{1}}\cdots x_{k}^{r_{k}}\,.$

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