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.)
+23247 Seat John first.
Then there are 8 students who can sit to the right of John, 7 students who can sit to the right of that student, 6 students who can sit to that student's right, and five students who can sit to that student's right.
Then seat Sam.
You can then seat one of the four remaining students to Sam's right, then one of the three remaining, one of the two remaining, and, finally, the last student.
There are 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 (or 8!) arrangements.
It doesn't make any difference if you choose to seat students to the right or to the left of John and it does not make any difference if you choose to seat Sam first.
