# Sum of Binomial Coefficients over Upper Index

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## 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:

$\dbinom 0 m + \dbinom 1 m + \dbinom 2 m + \cdots + \dbinom n m = \dbinom {n + 1} {m + 1}$

## Proof

 $\displaystyle \sum_{0 \mathop \le j \mathop \le n} \binom j m$ $=$ $\displaystyle \sum_{0 \mathop \le m + j \mathop \le n} \binom {m + j} m$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle \sum_{-m \mathop \le j \mathop < 0} \binom {m + j} m + \sum_{0 \mathop \le j \mathop \le n - m} \binom {m + j} m$ $\displaystyle$ $=$ $\displaystyle 0 + \sum_{0 \mathop \le \mathop j \mathop \le n - m} \binom {m + j} m$ Definition of Binomial Coefficient: negative lower index $\displaystyle$ $=$ $\displaystyle \binom {m + \left({n - m}\right) + 1} {m + 1}$ Rising Sum of Binomial Coefficients $\displaystyle$ $=$ $\displaystyle \binom {n + 1} {m + 1}$

$\blacksquare$