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:

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


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          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \binom {n + 1} {m + 1}\) \(\displaystyle \) \(\displaystyle \)                    

$\blacksquare$


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