Sum of Binomial Coefficients over Lower Index/Proof 3

Theorem

 * $\displaystyle \sum_{i \mathop = 0}^n \binom n i = 2^n$

where $\displaystyle \binom n i$ is a binomial coefficient.

Proof
From the Binomial Theorem, we have that:


 * $\displaystyle \forall n \in \Z_+: \left({x+y}\right)^n = \sum_{i \mathop = 0}^n \binom n i x^{n-i}y^i$

Putting $x = y = 1$ we get: