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.)
Fix one of the scientists at the head of table:
The other 4 can sit either to the left or to the right, or split 2 to the left and 2 to right, or 1 to right and 3 to the left or vise versa (it makes no difference) in this many ways:
4! = 24 ways
The remaining people (2 + 5 + 2 + 3 = 12) can be seated in:
12! =479,001,600 ways - since there is no restriction to their seating.
So, the total number of ways =24 x 479,001,600 = 11,496,038,400