Equation for binomial coefficients

Question

Let $n\in\Bbb N$, $k\in\{0,\dots,2n\}$. Is this equation valid? $$\binom{2n}{k}=\sum_{i=0,\dots,n\\ j=0,\dots,n\\i+j=k}\binom{n}{i}\binom{n}{j}$$

Answer 1

Yes, it holds. We have that $$\sum_{i=0,\dots,n\\ j=0,\dots,n\\i+j=k}\binom{n}{i}\binom{n}{j}=\sum_{i=0,\dots,n\\ j=0,\dots,n\\i+j=k}[x^i](1+x)^{n}\cdot [x^j](1+x)^{n}=[x^k](1+x)^{2n}=\binom{2n}{k}.$$