I came across this equation when solving another combinatorics problem. I needed to prove the following identity:
$$ \begin{aligned} \binom{n+1}{k+1}&=\sum_{i=m}^{n+m-k}{\binom{n-i}{k-m}\binom{i}{m}} \end{aligned} $$
My attempt:
The right hand side is selecting $k-m$ balls from a number of red balls and $m$ balls from a number of blue balls where the number of red and blue balls vary but the total is always $n$.
I imagine this is the same as lining up $n+1$ non - colored balls, then selecting $k+1$ balls from these. Then proceed to color all balls on the left side of the $(m+1)$th selected balls blue and on the right side of it red. Thus proving the equality.
My question is, does this equality or identity have a name? For research purposes.