25.2.4 ** Find the sum $S=\sum_{k=1}^{n} k^{2} \cdot \mathrm{C}_{n}^{k}$.
By analyzing $k \cdot \mathrm{C}_{n}^{k}-k \cdot \frac{n!}{k!(n-k)!}-n \cdot \frac{(n-1)!}{(k-1)!(n-k)!}-n \cdot \mathrm{C}_{n-1}^{k-1}$, we can obtain:
$$
\begin{aligned}
S & =n \cdot \sum_{k=1}^{n} \cdot k \mathrm{C}_{n-1}^{k-1}=n \cdot \sum_{k=1}^{n}(k-1) \mathrm{C}_{n-1}^{k-1}+n \cdot \sum_{k=1}^{n} \mathrm{C}_{n-1}^{k-1} \\
& -n \cdot(n-1) \sum_{k=2}^{n} \mathrm{C}_{n-2}^{k-2}+n \cdot 2^{n-1}-n \cdot(n-1) \cdot 2^{n-2}+n \cdot 2^{n-1} \\
& =n(n+1) \cdot 2^{n-2} .
\end{aligned}
$$