Example 4.12 Select $r(r \geqslant n)$ elements from an $n$-element set with repetition, where each element is chosen at least once. Find the number of such combinations with repetition.
Let the required value be $N$, then $N$ is the coefficient of $t^{r}$ in the expansion of
$$
A(t)=\left(t+t^{2}+t^{3}+\cdots\right)^{n}
$$

We have
$$
\begin{aligned}
A(t) & =t^{n}(1-t)^{-n}=t^{n} \sum_{k=0}^{\infty}\binom{n+k-1}{n-1} t^{k} \\
& =\sum_{k=0}^{\infty}\binom{n+k-1}{n-1} t^{n+k}=\sum_{r=n}^{\infty}\binom{r-1}{n-1} t^{r},
\end{aligned}
$$

Therefore,
$$
N=\binom{r-1}{n-1} \quad(r \geqslant n) .
$$