Sum of combinations with a condition

Question

Let $m,n,p,q,r$ be non-negative integers, with $0<m\leq n$ and $p+q+r=n$ The identity $\binom{n}{m}=\sum_{x+y+z=m}\binom{p}{x}*\binom{q}{y}*\binom{r}{z}$ holds? I already checked it for m=2, n=5.

Answer 1

The binomial coefficient $\binom{n}{m}$ counts the ways of selecting $m$ elements from the set $\{1,\cdots,n\}$.

Suppose we color the elements of $\{1,\cdots,n\}$ red, blue or green, the number of which are $p$, $q$ and $r$ respectively. Then the sum $\sum_{x+y+z=m}\binom{p}{x} \binom{q}{y} \binom{r}{z}$ counts the number of ways of drawing $m$ elements from the colored version of the set $\{1,\cdots,n\}$ and getting $x$ reds, $y$ blues and $z$ greens.

Since $(x,y,z)$ runs over all possibilities, this counts the total number of ways of drawing $m$ elements from the colored set $\{1,\cdots,n\}$. The colors don't affect the count though, so this is $\binom{n}{m}$.