Suppose I have a set $\{n_1, n_2, \cdots, n_m\}$ such that $n=\sum\limits_{i=1}^m n_i$. Is there a way to simplify the expression
$$ \prod\limits_{i=1}^m {n \choose n_i} $$
I feel like this might be related to multinomial coefficients but I am not sure.