# Identity of binomial coefficients: $\binom{n+1}{k+1}=\sum_{i=m}^{n+m-k}{\binom{n-i}{k-m}\binom{i}{m}}$

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.

• I’m not aware of any name for it, but it is a special case of (5.25) in Table $169$ of Graham, Knuth, & Patashnik, Concrete Mathematics. – Brian M. Scott Apr 13 at 5:02
• I would call it: "Vandermonde playing hockey". – Phicar Apr 13 at 21:45