Generalized Binomial and Multinomial Coefficients

In combinatorial analysis, the binomial coefficients \(\dbinom {n}{m}\) and the multinomial coefficients \(\dbinom {n}{n_1,\ldots ,n_q}\) are defined for any nonnegative integers \(0\leqslant m\leqslant n\) and any tuples \((n,n_1,\ldots ,n_q)\) of nonnegative integers satisfying \(q\geq 2\) and \(n_1+...+n_q=n\) by the relations \(\displaystyle \dbinom {n}{m}=\dfrac {n!}{m!(n-m)!}\quad \text{and}\quad \dbinom {n}{n_1,\ldots ,n_q}=\dfrac {n!}{n_1!...n_q!}. \)