Sum of Binomial Coefficients over Lower Index/Corollary
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Theorem
- $\ds \forall n \in \Z_{\ge 0}: \sum_{i \mathop \in \Z} \binom n i = 2^n$
where $\dbinom n i$ is a binomial coefficient.
Proof
From the definition of the binomial coefficient, when $i < 0$ and $i > n$ we have $\dbinom n i = 0$.
The result follows directly from Sum of Binomial Coefficients over Lower Index.
$\blacksquare$