At a meeting, five scientists, two mathematicians, five journalists, two biologists, and three politicians are to be seated around a circular table. How many different arrangements are possible if the scientists must all sit together (in five consecutive seats). (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)
We can treat the five scientists as a single "block" and arrange them first in a line, which can be done in 5! = 120 ways. We can then treat this block as a single entity and arrange the remaining 12 people (2 mathematicians, 5 journalists, 2 biologists, and 3 politicians) around the circular table. There are (12-1)! = 11! ways to arrange 12 distinct objects in a circle, but we must divide by 2! to account for the fact that the two mathematicians are indistinguishable from each other, and divide by 5! to account for the fact that the five scientists can be rotated within their block without changing the arrangement. Similarly, we must divide by 2! for the biologists and by 3! for the politicians. Therefore, the total number of distinct arrangements is:
5! x (11!/2!5!2!3!) = 120 x 8316 = 997,920.
Therefore, there are 997,920 different arrangements of the scientists, mathematicians, journalists, biologists, and politicians around the circular table if the five scientists must all sit together.