At a meeting, 5 scientists, 2 mathematicians, and a journalist are to be seated around a circular table. How many different arrangements are possible if the journalist must sit between two scientists. (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)

Answer 1

First, we will consider how many ways there are to seat the 5 scientists and 2 mathematicians around the circular table. We can treat the scientists and mathematicians as distinguishable objects, since there are no restrictions on how they are seated. There are 7 people in total, so there are $7!$ ways to arrange them around a circular table.

Now, we need to consider the restriction that the journalist must sit between two scientists. We will consider the journalist and the two scientists as a single group, which means we effectively have 6 objects to arrange (the group, the remaining 3 scientists, and the 2 mathematicians).

Considering the group as a single object, we can arrange the 6 objects around the circular table in $(6-1)! = 5!$ ways. However, since the group can rotate, we need to divide this number by 6 to account for different circular arrangements of the group. Therefore, there are $\frac{5!}{6}$ ways to arrange the objects.

Putting it all together, the total number of different seatings is $\frac{7!}{\frac{5!}{6}} = \boxed{84}$.