I came across the following identity while computing certain expressions: $$\sum_{i=0}^m \binom{l}{i}\binom{m+n-l}{m-i} = \binom{m+n}{m}.$$ Here, $m,n,l \geq 0$ are fixed and $l \leq m+n$. Also assume that $\binom{x}{y} = 0$ whenever $x < y$.
I verified this identity for small values of $m$, $n$ and $l$, and I think I have come up with a combinatorial argument for why this identity is true. Can someone please verify it for me?
Proof.
The RHS is the number of ways of choosing $m$ objects out of a collection of $m+n$ distinct objects.
We can make this selection in the following way as well. Divide the collection into two distinct bunches of $l$ objects and $m+n-l$ objects. We can select $m$ objects out of our $m+n$ objects by selecting $i$ objects from the $l$-bunch and the remaining $m-i$ from the $(m+n-l)$-bunch, for each possible value of $i$ between $0$ and $m$ (in the sense that we don't try to select more than $l$ objects out of our first bunch, or more than $m+n-l$ objects out of our second bunch). But, this is nothing but the expression on the LHS.
Is my proof valid? I would be grateful if someone can provide alternative proofs of this identity as well. Thanks in advance!