I've been trying to think up of two combinatorial proofs but I'm confused on the first and just completely stuck on where to begin for the second.
$$\sum_{k=-m}^n \binom{m+k}{r}\binom{n-k}{s} = \binom{m+n+1}{r+s+1}$$ For the RHS, I understand that it is basically looking at the number of $(r+s+1)$ subsets that can be formed from $(m+n+1)$. On expanding the LHS to yield $\binom{0}{r}\binom{n+m}{s}+\binom{1}{r}\binom{n+m-1}{s}...\binom{m}{r}\binom{n}{s}+\binom{m+1}{r}\binom{n-1}{s}+...\binom{m+n}{r}\binom{0}{s}$, I see that it is basically adding together the various ways of drawing a constant $r$ and $s$ from two changing populations that sum up to $m+n$. However, I don't actually see where the $+1$ comes from. Is there anything wrong with my intepretation of the LHS?
$$\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$$ I know that RHS is counting the number of subsets of 3 that can be formed from $n$. However, I do not have any idea of how to be interpreting the LHS.