Question: Let (S(n)=sum_{k=0}^{n}left(begin{array}{l}n kend{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
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 \(a=b=1\).
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