combinatorics - Generalized Vandermonde's identity

Can you please provide a reference to the following generalization of Vandermonde's identity?

Given a positive integer $k$ and nonnegative integers $n_1, n_2, \ldots, n_k$ and $m$, it holds that $$\sum_{i_1+i_2+\cdots+i_k=m} \binom{n_1}{i_1} \binom{n_2}{i_2} \cdots \binom{n_k}{i_k} = \binom{n_1+n_2+\cdots+n_k}{m}.$$ The proof is well-known and based on the idea of counting in two different ways the coefficient of $x^m$ in the polynomial $(1+x)^{n_1+n_2+\cdots+n_k}$.