At a meeting, five scientists, two mathematicians, and a journalist 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) and the mathematicians must sit next to each other? (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)
First, we can treat the scientists as a single unit, so we have 3 units (the scientists, the mathematicians, and the journalist) to arrange around the table. This can be done in (3−1)!=2!=2 ways.
Next, we need to arrange the scientists and the mathematicians within their respective units. The scientists can be arranged in 5! ways, and the mathematicians can be arranged in 2! ways.
So the total number of arrangements is 2×5!×2!=240.