Sum of Binomial Coefficients over Upper Index

Theorem
Let $m \in \Z$ be an integer such that $m \ge 0$.

Then:
 * $\displaystyle \sum_{j \mathop = 0}^n \binom j m = \binom {n+1} {m+1}$

where $\displaystyle \binom j m$ denotes a binomial coefficient.

That is:
 * $\displaystyle \binom 0 m + \binom 1 m + \binom 2 m + \cdots + \binom n m = \binom {n+1} {m+1}$