In how many ways can we seat 6 people around a round table if fred and gwen insist on sitting opposite each other? (two seatings are considered equivalent if one is a rotation of the other.)

Make a picture of the problem.

Fred and Gwen are seated opposite each other, and we name the 4 remaining seats 1, 2, 3, 4, starting from the right of Fred.

There are 4 options for seat 1,

then 3 for the seat 2, after one is seated in the 1.st seat

2 options for the seat 3,

and 1 for the seat 4.

That is, the 4 friends can sit in 4*3*2*1=24 different ways.

24