Binomial Theorem and number of subsets

Let S be a set of $n$ objects; then the binomial coefficient $(^n_k)$ is the number of  $k$-elements subsets of $S$. Thus $\sum_{k=0}^n(^n_k)$ is the number of subsets of $S$ of all possible sizes from 0 through $n$.

The binomial theorem says that

\[(x+y)^n=\sum_{k=0}^n(^n_k)x^ky^{n-k}\quad;(1)\]

if you substitue $x=y=1$ in $(1)$ , you get

\[(1+1)^n=\sum_{k=0}^n(^n_k)1^k1^{n-k}=\sum_{k=0}^n(^n_k) \quad; (2)\]

And of course$(1+1)^n=2^n$, so $(2)$ reduces to

\[2^n=\sum_{k=0}^n(^n_k)={number\;of\;subsets\;of\;S}.\]