Example 6 (1) 8 boys and 25 girls are to be arranged in a line, with at least 2 girls between any two boys. How many different arrangements are there?
(2) 8 boys and 25 girls are to be arranged in a circle, with at least 2 girls between any two boys. How many different arrangements are there?
Solution: (1) Let the symbol “*” represent a boy standing at the position, and the symbol “…” represent several girls standing at the position, then we have $\cdots * \cdots * \cdots * \cdots * \cdots * \cdots * \cdots * \cdots * \cdots * \cdots$,
9 “…” positions from left to right stand $x_{1}, x_{2}, x_{3}, x_{4}$, $x_{5}, x_{6}, x_{7}, x_{8}, x_{9}$ girls respectively, so $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}=25$, and $x_{1} \geqslant 0, x_{i} \geqslant 2 (i=2,3,4, \cdots \cdots, 8), x_{9} \geqslant 0$. Therefore, the number of girls in each part is in a $1-1$ correspondence with the integer solutions of the above equation that meet the requirements. The number of integer solutions of the above equation that meet the requirements is $C_{19}^{8}$ (for the method of finding, refer to Example 4), so there are $C_{19}^{8}$ ways to arrange the number of girls in each part. Then, arrange the 25 girls in a full permutation and arrange the girls according to the number of girls in each part, while also arranging the 8 boys in a full permutation, so there are $C_{19}^{8} \times P_{25}^{25} \times P_{8}^{8}$ ways to arrange 8 boys and 25 girls that meet the requirements.
(2) As shown in Figure 30-2, the “*” on the circle represents the position where a boy stands. Considering from $P$ in a clockwise direction, let $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}$ girls stand between two adjacent boys, then $x_{1}+x_{2}+x_{3}+\cdots \cdots+x_{8}=25$, and $x_{i} \geqslant 2, (i=1,2,3, \cdots \cdots, 8)$. The allocation scheme of the number of girls in each part is in a $1-1$ correspondence with the integer solutions of the above equation that meet the conditions, so the number of allocation schemes of the number of girls is equal to the number of integer solutions of the above equation that meet the requirements. The number of integer solutions of the above equation that meet the requirements is $C_{16}^{7}$. Therefore, the number of circular permutations of the boys is $P_{7}^{7}$, and arranging the girls in a full permutation according to the allocation scheme, the number of different permutations that meet the requirements is $C_{16}^{7} P_{7}^{7} P_{25}^{25}=\frac{25!\cdot 16!}{9!}$.