【Example 3】A party needs to arrange 10 singing programs and 4 dance programs, hoping that at least two singing programs are inserted between every two dances. How many different performance orders can be arranged?
To solve this, we can first consider arranging the 4 dance programs, which requires selecting a limited-distance combination from $\{1,2, \cdots, 14\}$. Since $n=14, k=4, m=3$, we know there are $C_{14-3 \times 2}^{4}=C_{8}^{4}$ ways. Then, the dance and singing programs are arranged in their respective performance orders, with 4! and 10! ways respectively. Therefore, the total number of different performance orders is
$$
C_{8}^{4} \cdot 4!\cdot 10!=\frac{8!\cdot 10!}{4!}
$$