Let \(S(n)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}ight)\).
(a) Use Pascal's Triangle to compute \(S(n)\) for \(n=1,2,3,4\).
(b) Prove that \(S(n)=2^{n}\) for all \(n \geq 1\). Expand \((a+b)^{n}\) and evaluate at \(a=b=1\).