Prove the following summation of binomial coefficients. \sum_{i=0}^n \sum_{j=0}^m {i+j \choose j} = {n + m +2 \choose n+1} – 1.

First, by Pascal’s rule that {i+j \choose j} = {i+j-1 \choose j} + {i+j-1 \choose j-1} , …

A Double Sum of Binomials Read more »