Binomial coefficient identity proof

Question

I'm struggling with proving the identity $$\sum_{k=p}^{q}\binom{l}{m+k}\binom{s}{n+k}=\binom{l+s}{l-m+n}$$ where $$p=-\min(m,n)~ \text{and}~q=\min(l-m,s-q).$$ It reminds me of Vandermonde's identity but still I can't get it right. I would appreciate an algebraic or combinatorial proof.

Answer 1

$\sum_{k=p}^{q}{\left(\binom{l}{m+k}\binom{s}{n+k}\right)}$ is the coefficient of $x^{m+k}\cdot \left(\frac{1}{x}\right)^{n+k}=x^{m-n}$ from $\left(1+x\right)^{l}\left(1+\frac{1}{x}\right)^{s}=\frac{\left(1+x\right)^{l+s}}{x^{s}}$ which is $\binom{l+s}{m-n+s}=\binom{l+s}{l-m+n}.$ Here $p=-min(m,n)$ and $q=min(l-m,s-n).$