In how many ways may we string n distinct beads on a necklace without a clasp? (Perhaps we make the necklace by stringing the beads on a string, and then carefully gluing the two ends of the string together so that the joint can't be seen. Assume someone can pick up the necklace, move it around in space and put it back down, giving an apparently different way of stringing the beads that is equivalent to the first.)

Hint 1: How could we get a list of beads from a necklace?

Hint 2: When we cut the necklace and string it out on a table, there are 2n lists of beads we could get. Why is it2n2nrather thann?