Combinatorics – Algebraic Proof of Combinatorial Identity

Question

I would like to obtain the algebraic proof for the following identity. I already know the combinatorial proof but the algebraic proof is evading me.

$$\sum_{r=0}^n\binom{n}{r}\binom{2n}{n-r}=\binom{3n}{n}$$

Thanks.

Answer 1

Hint

Calculate by two ways the coefficient of the term $x^n$ of $(1+x)^n(1+x)^{2n}$ .