The binomial theorem states:
\[\begin{align} (x+y)^n &= \binom n 0 x^n + \binom n 1 x^{n-1} y^1 + \binom n 2 x^{n-2} y^2 + \cdots \\ &= \sum^n_{i=0} \binom n i x^{n-i} y^i \end{align}\]
Let \(x=1\) and \(y=1\):
\[\begin{align} 2^n &= \sum^n_{i=0} \binom n i 1^{n-i} 1^i = \sum^n_{i=0} \binom n i \\ &= \binom n 0 + \binom n 1 \cdots + \binom n {n-1} + \binom n n \end{align} \]