George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8,and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of each seating arrangement are not considered different?
The wife's position automatically determines the husband's position. There are 4! = 24 ways to order the couples, but there are 4 ways to rotate them, and 2 ways to reflect them. So the numbe of arrangements is 24/(4*2) = 3.