Example 3 Find the value of $\sum_{k=1}^{n} k^{2} \mathrm{C}_{n}^{k}$.
(23rd Putnam Mathematical Competition)
$$
\begin{array}{l}
\text { Solution: } \sum_{k=1}^{n} k^{2} \mathrm{C}_{n}^{k}=\sum_{k=1}^{n} k \cdot k \mathrm{C}_{n}^{k}=\sum_{k=1}^{n} k \cdot n \mathrm{C}_{n-1}^{k-1} \\
=n \sum_{k=1}^{n}[(k-1)+1] \mathrm{C}_{n-1}^{k-1} \\
=n \sum_{k=1}^{n} \mathrm{C}_{n-1}^{k-1}+n \sum_{k=2}^{n}(n-1) \mathrm{C}_{n-2}^{k-2} \\
=n \cdot 2^{n-1}+n(n-1) \cdot 2^{n-2} \\
=n(n+1) \cdot 2^{n-2} .
\end{array}
$$

Note. For such combinatorial sum problems, the property $k \mathrm{C}_{n}^{k}=n \mathrm{C}_{n-1}^{k-1}$ is often used to transform each term in the expression, and then the problem is solved using properties of binomial coefficients.
$$