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.)

I do not understand this!

Similar to your previous question.

Whatever place John or Sam are in, the other one must be in position #6 over from him.

let's line them up in a straight line,
the first spot can be filled in 2 ways, (John or Sam)
then the 6th spot can be filled in 1 way (one of John or Sam)
now fill in the other 8 places, so I see them lined up as follows:
2x8x7x6x5x1x4x3x2x1
= 80640

So when placing them around the table, we can consider the first spot to be any of on 10 places, so

number of ways 80640/10 = 8064

To understand the problem, let's break it down step by step.

We have a circle with 10 seats, and John and Sam want to be seated diametrically opposite each other. This means that once John's seat is chosen, Sam's seat is automatically determined.

To solve this problem, we first need to determine the number of ways to arrange the remaining 8 kids in the circle. Once we have that, we can multiply it by 2 (since John and Sam can switch places) to get the final answer.

To calculate the number of ways to arrange 8 kids in a circle, we can use the concept of permutations. In a circle, the starting point is not fixed, so we need to divide the total number of arrangements by the number of rotations (which is the number of kids in this case).

The formula to calculate the number of ways to arrange n objects in a circle is (n-1)!

In this case, there are 8 kids remaining, so the number of ways to arrange them is (8-1)! = 7!.

Now, multiplying by 2 to account for John and Sam switching places, we get the final answer:

Answer = 2 * 7!

To calculate this further, we can evaluate:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040

Therefore:

Answer = 2 * 5040 = 10080

So, there are 10,080 ways to seat the 10 kids in a circle if John and Sam insist on being seated diametrically opposite each other.