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:

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

Proof

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \sum_{0 \mathop \le j \mathop \le n} \binom j m$$ $$=$$ $$\displaystyle$$ $$\displaystyle \sum_{0 \mathop \le m + j \mathop \le n} \binom {m + j} m$$ $$\displaystyle$$ $$\displaystyle$$ Permutation of Indices $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \sum_{-m \mathop \le j \mathop < 0} \binom {m + j} m + \sum_{0 \mathop \le j \mathop \le n - m} \binom {m + j} m$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle 0 + \sum_{0 \mathop \le \mathop j \mathop \le n - m} \binom {m + j} m$$ $$\displaystyle$$ $$\displaystyle$$ Definition of binomial coefficient: negative lower index $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \binom {m + \left({n - m}\right) + 1} {m + 1}$$ $$\displaystyle$$ $$\displaystyle$$ Rising Sum of Binomial Coefficients $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \binom {n + 1} {m + 1}$$ $$\displaystyle$$ $$\displaystyle$$

$\blacksquare$