Sum of Binomial Coefficients over Lower Index/Corollary

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.

$\blacksquare$