3. Draw $m$ times with replacement from $1,2, \cdots, n$. Find the probability that the sum of the $m$ numbers drawn is $k$.


Let the random variable $\xi_{i}$ denote the number obtained in the $i$-th draw. Then the sum of $m$ numbers is
$$
\xi=\xi_{1}+\xi_{2}+\cdots+\xi_{m} \text {. }
$$

Thus, the generating function of $\xi$ is
$$
\varphi(x)=\left(\sum_{i=1}^{n} \frac{x^{i}}{n}\right)^{m}=\frac{1}{n^{m}}\left(\sum_{i=1}^{n} x^{i}\right)^{m} .
$$

The probability $P(\xi=k)$ is the coefficient of $x^{k}$ in the expansion of the generating function.