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# Multinomial coefficients

${\displaystyle (n_{1},\,n_{2},\,\ldots ,\,n_{k})!:={\frac {(n_{1}+n_{2}+\cdots +n_{k})!}{n_{1}!\,n_{2}!\,\cdots \,n_{k}!}},\quad k\geq 1,\,}$
where ${\displaystyle n!}$ is a factorial, are a generalization of the binomial coefficients for the ${\displaystyle k}$-multinomials (binomials being 2-multinomials)
${\displaystyle (x_{1}+x_{2}+\cdots +x_{k})^{n}=\sum _{{n_{1},\,n_{2},\,\ldots ,\,n_{k}\geq 0} \atop {n_{1}+n_{2}+\cdots +n_{k}=n}}(n_{1},\,n_{2},\,\ldots ,\,n_{k})!\ {x_{1}}^{n_{1}}\,{x_{2}}^{n_{2}}\,\cdots \,{x_{k}}^{n_{k}},\quad k\geq 1.\,}$
For the degenerate case ${\displaystyle k=1}$, the result is always 1.