Generalized Vandermonde's identity

Statement

One can generalize Vandermonde's identity as follows:

$$\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }.$$

This identity can be obtained through the algebraic derivation above when more than two polynomials are used, or through a simple double counting argument.

On the one hand, one chooses $\textstyle k_1$ elements out of a first set of $\textstyle n_1$ elements; then $\textstyle k_2$ out of another set, and so on, through $\textstyle p$ such sets, until a total of $\textstyle m$ elements have been chosen from the $\textstyle p$ sets. One therefore chooses $\textstyle m$ elements out of $\textstyle n_1+\dots +n_p$ in the left-hand side, which is also exactly what is done in the right-hand side.