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The **multinomial coefficients**

- $(n_{1},\,n_{2},\,\ldots ,\,n_{k})!:={\frac {(n_{1}+n_{2}+\cdots +n_{k})!}{n_{1}!\,n_{2}!\,\cdots \,n_{k}!}},\quad k\geq 1,\,$

where $n!$ is a factorial, are a generalization of the binomial coefficients for the $k$-multinomials (binomials being 2-multinomials)

- $(x_{1}+x_{2}+\cdots +x_{k})^{n}=\sum _{{n_{1},\,n_{2},\,\ldots ,\,n_{k}\geq 0} \atop {n_{1}+n_{2}+\cdots +n_{k}=n}}(n_{1},\,n_{2},\,\ldots ,\,n_{k})!\ {x_{1}}^{n_{1}}\,{x_{2}}^{n_{2}}\,\cdots \,{x_{k}}^{n_{k}},\quad k\geq 1.\,$

For the degenerate case $k=1$, the result is always 1.