1.38 Find the number of combinations of $k$ numbers $\left\{j_{1}\right.$, $\left.j_{2}, \cdots, j_{k}\right\}$ chosen from the set $\{1,2, \cdots, n\}$ that satisfy the following conditions:
(1) $1 \leqslant j_{1}<j_{2}<\cdots<j_{k} \leqslant n$;
(2) $j_{2}-j_{1} \geqslant m+1$, where $m>1$ is a fixed positive integer;
(3) There exists $h_{0}, 1 \leqslant h_{0} \leqslant k-1$, such that $j_{h_{0}+1}-j_{h_{0}} \geqslant m+1$.
[Solution] From the result of problem 36, we know that the number of $k$-tuples satisfying conditions (1) and (2) is $C_{n-(k-1)(m-1)}^{k}$. These tuples that do not satisfy condition (3) must make the equality in (2) hold, i.e., these $k$ numbers form an arithmetic sequence with a common difference of $m$. Clearly, such a tuple is uniquely determined by its first term. In $\{1,2, \cdots, n\}$, the numbers that can serve as the first term of such a sequence are exactly $n-(k-1) m$ in number. Therefore, the number of $k$-tuples that satisfy conditions (1) and (2) but not condition (3) is $n-(k-1) m$. Hence, the number of $k$-tuples we are looking for is $C_{n-(k-1)(m-1)}^{k} - n + (k-1) m$.