Example 7 If elements are allowed to be selected repeatedly, find the number of combinations of selecting $r$ elements from $n$ different elements.
Solution: Number these $n$ elements from 1 to $n$, so the selection of any $r$ elements becomes a combination of $r$ natural numbers $\left(a_{1}, a_{2} \cdots \cdots, a_{r}\right)$, where we can assume $1 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \cdots a_{r} \leqslant n$. Let $b_{i}=a_{i}+i-1, i=1,2, \cdots \cdots r$, then $1 \leqslant b_{1}<b_{2} \cdots \cdots <b_{r} \leqslant n+r-1$. The set $\left\{b_{1}, b_{2} \cdots \cdots, b_{r}\right\}$ is a combination of $r$ different elements from the set $\{1,2,3, \cdots \cdots, n+r-1\}$ of $n+r-1$ elements. Conversely, for any set $\left\{b_{1}, b_{2}, \cdots \cdots, b_{r}\right\}$, since $1 \leqslant b_{1}<b_{2}<\cdots \cdots<b_{r} \leqslant n+r-1$, and $a_{i}=b_{i}-i+1$, then $1 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \cdots \leqslant a_{r} \leqslant n$, meaning $\left(a_{1}, a_{2}, \cdots \cdots, a_{r}\right)$ is a combination of $r$ elements (with repetition allowed) selected from $\{1,2, \cdots \cdots, n\}$. Therefore, these two types of combinations can be put into a 1-1 correspondence, so the required number of combinations is $C_{n+r-1}^{r}$.