Show that $\sum_{k=0}^{\infty} \binom{r}{k} \binom{s}{n+k} = \binom{r+s}{r+n}$. I can't resolve this exercise and I need some tips.
Let be $n$ a integer, $s$ a real number and $r \geq 0$ a integer. Show that
$$ \sum_{k=0}^{\infty} \binom{r}{k} \binom{s}{n+k} = \binom{r+s}{r+n} $$

Both sides are polynomials in $s$ (the terms of the series are zero when $k>r$), so it suffices to show they are equal when $s$ is a nonnegative integer.
Suppose that I have $s$ statesmen and $r$ robots.  ${{s+r}\choose{n+r}}$ counts the number of ways that we can form a caucus with $n+r$ members.
But if our caucus has $n+k$ statesmen ($0\leq k \leq r$), then we must exclude exactly $k$ robots.  So the number of such caucuses is ${r\choose k}{s \choose {n+k}}$.
