Define the binomial coefficient $\left(\begin{array}{c}n \\ m\end{array}\right)$. Prove directly from your definition that

$$(1+z)^{n}=\sum_{m=0}^{n}\left(\begin{array}{c} n \\ m \end{array}\right) z^{m}$$

for any complex number $z$.

(a) Using this formula, or otherwise, show that

$$\sum_{k=0}^{3 n}(-3)^{k}\left(\begin{array}{l} 6 n \\ 2 k \end{array}\right)=2^{6 n}$$

(b) By differentiating, or otherwise, evaluate $\sum_{m=0}^{n} m\left(\begin{array}{c}n \\ m\end{array}\right)$.

Let $S_{r}(n)=\sum_{m=0}^{n}(-1)^{m} m^{r}\left(\begin{array}{c}n \\ m\end{array}\right)$, where $r$ is a non-negative integer. Show that $S_{r}(n)=0$ for $r<n$. Evaluate $S_{n}(n)$.