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}.]