Let $X \sim \text{Bin}(m,p)$ and $Y \sim \text{Bin}(n,p)$. I want to find the distribution of $W = X + Y$ using a convolution theorem that says: $$f_W(w) = \sum\limits_x f_X (x) f_Y(w-x).$$ So using the binomial distributions for $X$ and $Y$: $$\Rightarrow \sum\limits_{x=0}^w {m \choose x} p^x(1-p)^{m-x} {n \choose w-x}p^{w-x}(1-p)^{n-w+x}$$
$$\Rightarrow \sum\limits_{x=0}^w {m \choose x} {n \choose w-x} p^w(1-p)^{m+n-w}.$$
And it's the sum over $x$, so:
$$ \Rightarrow p^w(1-p)^{m+n-w} \sum\limits_{x=0}^w {m \choose x} {n \choose w-x}.$$
My question is, what is the best approach to evaluate $ \sum\limits_{x=0}^w {m \choose x} {n \choose w-x}$?? Kinda stuck on this part, but I know it should equal ${m+n \choose w}$ somehow...