$\displaystyle 2^n = \sum_{r=0}^r \begin{pmatrix} n\\ r \end{pmatrix} $ using the Binomial Theorem
This is a standard theorem if we know that $\displaystyle \left( {a + b} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}a^k b^{n - k} }$
Now let $\displaystyle a=1~\&~b=1$