1.598 girls and 25 boys stand in a circle, with at least two boys standing between any two girls. How many different arrangements are there (arrangements that can be made to coincide by rotating the circle are considered the same)?
[Solution] Fix one of the girls and denote her as $A$. For any circular arrangement that meets the requirement, transform it into a linear arrangement starting from $A$ in a clockwise direction. Since each girl is followed by at least two boys, we can associate each girl along with the two boys immediately following her with a black ball, and each of the remaining 9 boys with a white ball. When we only consider the differences between boys and girls but not the differences among girls or among boys, this correspondence is a bijection, and the correspondence between the resulting arrangements is also a bijection. Thus, the problem is reduced to calculating the number of all different arrangements of 8 black balls and 9 white balls where the first ball is black. Clearly, the number of different arrangements in this case is $C_{16}^{7}$.

For each arrangement of black and white balls, each black ball corresponds to a group of three people, where the first is a girl and the next two are boys. Arrange the 7 girls other than $A$ into the last 7 groups, which can be done in 7! ways; arrange the 25 boys into 25 positions, which can be done in 25! ways. By the multiplication principle, the total number of different arrangements is $\frac{16!25!}{9!}$.