Alternating Sum and Difference of Binomial Coefficients for Given n/Proof 1

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
We note:
 * $\displaystyle \binom n 0 = \binom {n-1} 0 = 1$

so:
 * $\displaystyle \binom n 0 - \binom {n-1} 0 = 0$


 * $\displaystyle \left({-1}\right)^{n-1} \binom {n-1} {n-1} = - \left({-1}\right)^n \binom n n = - \left({-1}\right)^n$

so:
 * $\displaystyle \left({-1}\right)^{n-1} \binom {n-1} {n-1} + \left({-1}\right)^n \binom n n = 0$

Hence the result.