The Smith College Diploma Problem

At Smith College, the graduation exercises traditionally proceed as follows: Although each diploma is made out to a particular girl, all the diplomas are initially given out at random. All of the girls who do not get their own diplomas then form a circle, and each passes the diploma she has to the girl on her right. Those who now have their own diplomas drop out, and the remaining girls again pass their diplomas to the right, and so on. This procedure is repeated until each girl has her own diploma. If there are n girls in the graduating class what is the probability that it takes precisely k passes before each girl has her own diploma?