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.)
There are a total of 17 people attending the meeting, so if the scientists are seated together in five consecutive seats, the remaining 12 people can be seated in the other 12 seats.
The number of ways to seat the scientists is just 1 (since they must all sit together), and the number of ways to seat the remaining 12 people is simply 12! (the number of permutations of 12 items).
However, we must divide by 2 to account for rotations of the table, since two seatings are considered equivalent if one can be obtained from rotating the other.
Therefore, the total number of different arrangements is:
1 * 12! / 2 = 72576
So there are 72,576 different arrangements possible.