Alternating Sum and Difference of Binomial Coefficients for Given n/Proof 2

Theorem

 * $\displaystyle \sum_{i \mathop = 0}^n \left({-1}\right)^i \binom n i = 0$ for all $n \in \Z: n > 0$

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

Proof
From the Binomial Theorem, we have that:


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

Putting $x = 1, y = -1$, we get: