Example 1. Select 3 elements $i, j, k$ from the set $\{1,2, \cdots, n\}$, such that (i) $1 \leqslant i \leqslant j \leqslant k \leqslant n$, (ii) $1 \leqslant i<j<k \leqslant n$, and $j-i \geqslant m, k-j \geqslant m$. How many different ways are there to select them?
(i) Establish a correspondence: $(i, j, k) \xrightarrow{\mathrm{f}}\left(i^{\prime}, j^{\prime}, k^{\prime}\right)$, such that $i^{\prime}=i, j^{\prime}=j+1, k^{\prime}=k+2$, and $1 \leqslant i^{\prime} < j^{\prime} < k^{\prime} \leqslant n+2$. This is a one-to-one correspondence, so the original problem is transformed into selecting three distinct numbers from $\{1,2, \cdots, n+2\}$, which has $C_{\mathrm{n}+2}^{3}$ ways of selection.
(ii) Similarly to (i), establish a one-to-one correspondence $(i, j, k) \xrightarrow{\mathrm{r}}\{i, j-(m-1), k-2(m-1)\}$. The problem is transformed into selecting three different elements from $\{1,2, \cdots, n-2(m-1)\}$, which has $C_{\mathrm{n}-2(m-1)}^{3}$ ways of selection.