A binomial identité

Question

I tried to solve this exercise but i failed. If anyone can give me hint or solution i would be grateful.

Let $(n,p,q)\in\mathbb{N}^{3}$ such that $n\leq p$ and $n\leq q$. Show that:$$\sum_{k=0}^{n}\binom{p}{k}\binom{q}{n-k}=\binom{p+q}{n}.$$

Answer 1

Since $$ (1+x)^p(1+x)^q=(1+x)^{p+q} $$ or $$ \sum_{k=0}^p\binom{p}{k}x^k\sum_{i=0}^q\binom{q}{i}x^i=\sum_{i=0}^{p+q}\binom{p+q}{i}x^i $$ comparing the coefficients of $x^n$ for both sides, you will have $$\sum_{k=0}^{n}\binom{p}{k}\binom{q}{n-k}=\binom{p+q}{n}.$$