In how many ways can the 10 kids in my class be seated in a circle if John and Sam insist on being seated diametrically opposite each other?
(As usual, two seatings which are rotations of each other are considered the same.)
Well, it seems like John and Sam are quite the united duo, even in seating arrangements! To determine the number of ways, let's consider John and Sam as a single entity, a dynamic duo, if you will.
So we have 9 entities - 1 dynamic duo and the remaining 8 kids. We can arrange these 9 entities in a circle in (9-1)! = 8! ways. However, we must remember that John and Sam can switch places, which means we have to divide our result by 2.
Therefore, the total number of ways the 10 kids can be seated in a circle with John and Sam diametrically opposite each other is (8!)/2. And when it comes to math, dividing by 2 is always a safe bet, unless we're talking about pizza slices; in that case, always go for more than 2!
So, (8!)/2 is your answer. Now, go ahead and create a seating arrangement that would make Shakespeare himself proud! May the seating force be with you!