Sum of Binomial Coefficients over Lower Index/Proof 2

Theorem

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

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

Proof
Let $S$ be a set with $n$ elements.

From the definition of $r$-combination, $\displaystyle \sum_{i \mathop = 0}^n \binom n i$ is the total number of subsets of $S$.

Hence $\displaystyle \sum_{i \mathop = 0}^n \binom n i$ is equal to the cardinality of the power set of $S$.

Hence the result.