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.)
You have 17 people.
Seat one biologist at the head of table. That leaves 16 people. 5 of them are scientists who want to sit together. We will consider them as 1 person. So, we have:16 - 5 + 1=12 people who can be seated in:
12! ways. But, the 5 scientists can rearrange themselves in 5! ways. So, the total seating of all 17 people should be: 12! x 5! =57,480,192,000 ways of seating 17 people.