Four mathematicians, three physicists, and an engineer are to be seated equally spaced around a circular table. How many different arrangements are possible if the mathematicians must all sit together (in four consecutive seats) and the physicists must all sit together (in three consecutive seats)? (As usual, two arrangements are identical if one is a rotation of the other.)
First, we can choose the seats for the four mathematicians. Then we seat the physicits, then the engineer. There are 4! ways to seat the mathematicians, then 3! ways to seat the physicists, then 1! ways to seat the engineer. But then we must divide by 8, because we can rotate the table, so there are 8!*4!*3!/8 = 725760 ways to seat everyone.