Sum of Binomial Coefficients over Upper Index/Proof 2

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Theorem

\(\ds \sum_{j \mathop = 0}^n \binom j m\) \(=\) \(\ds \binom {n + 1} {m + 1}\)
\(\ds \) \(=\) \(\ds \dbinom 0 m + \dbinom 1 m + \dbinom 2 m + \cdots + \dbinom n m = \dbinom {n + 1} {m + 1}\)


Proof

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

$\blacksquare$


Sources