For exam review, I did this question:
"How many ways can 6 keys be placed on a circular key ring? Both sides of the ring are the same, and here is no way to tell which is the "first" key on the ring."
My first thought is P(6,6)/6 = 120. The idea being that each permutation "matches" five other permutations when rotated around the ring. However, the book answer is 60, not 120, so evidently there is something off in my thinking.