Example 3 Find $S=\sum_{k=0}^{n} \frac{(\dot{m}+k)!}{k!}$.
(1962, Beijing Mathematical Competition (Grade 12))
\begin{aligned} \text { Sol } & : S=\sum_{k=0}^{n} \frac{(m+k)!}{k!} \\ = & m!\left[1+\frac{m+1}{1!}+\frac{(m+1)(m+2)}{2!}+\cdots\right. \\ & \left.+\frac{(m+1)(m+2) \cdots(m+n)}{n!}\right] \\ = & m!\left(\mathrm{C}_{m}^{0}+\mathrm{C}_{m+1}^{1}+\mathrm{C}_{m+2}^{2}+\cdots+\mathrm{C}_{m+n}^{n}\right) \\ = & m!\left(\mathrm{C}_{m+1}^{m+1}+\mathrm{C}_{m+1}^{m}+\mathrm{C}_{m+2}^{m}+\cdots+\mathrm{C}_{m+n}^{m}\right) \\ = & m!\mathrm{C}_{m+n+1}^{m+1}=\frac{(n+m+1)!}{n!(m+1)} .\end{aligned}