I wanted to find the value of the following expression
$$\sum_{r=0}^{20} (-1)^r. \binom {30}{r}. \binom {30}{r+10}$$
The Vandermonde's identity could have been used in this problem but the $(-1)^r$ makes it of no use. Using the binomial expansion and some brute force I got the answer as the coefficient of $x^{20}$ in the expansion $(1-x^2)^{30}$. But I am not able to provide the proof in writing because it was just by brute force and some observation. Can someone please provide a hint.