Cardinality of Power Set of Finite Set/Proof 2

Proof
Enumerating the subsets of $S$ is equivalent to counting all of the ways of selecting $k$ out of the $n$ elements of $S$ with $k = 0, 1, \ldots, n$.

So, from Cardinality of Set of Subsets, the number we are looking for is:


 * $\ds \card {\powerset S} = \sum_{k \mathop = 0}^n \binom n k$

But from the binomial theorem:


 * $\ds \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$

It follows that:
 * $2^n = \ds \paren {1 + 1}^n = \sum_{k \mathop = 0}^n \binom n k \paren 1^{n - k} \paren 1^k = \sum_{k \mathop = 0}^n \binom n k = \card {\powerset S}$