Here is the problem:
$\displaystyle\sum_{k=0}^{p}\binom{p+q-k}{q}\binom{r+k}{r}\ = \binom{p+q+r+1}{p}$
How to prove this using combinatorics? I don't want to use algebra or something. My idea was that since we are not explicitly choosing $k$, we probably shouldn't choose from a subset which can be added to $r$ or subtracted from $p+q$,instead we can imagine a bit string of probably p+q+r+1 length with $r+1$ ones and $p+q$ zeroes and consider counting on basis of occurrence of last one. But it isn't working out to be correct. Can someone please help