Question

In how many ways can 4 boys and 4 girls sit around a circle table if all the boys sit together? (Rotations of the same arrangement are still considered the same. Each boy and girl is unique, not interchangeable.)

Answer 1

There are 4! ways to sit the boys, and 4! ways to sit the girls.
Whether the boys sit to the left or the right of the girls don't matter, since it is a circular table AND rotations are not considered distinct arrangements.
So the total number of arrangements is the product of the two, namely n=4!4!.   (note: (4!)^2 or (four factorial) squared.