Exercise 5.2.1

In HIRE-ASSISTANT, assuming that the candidates are presented in a random order, what is the probability that you hire exactly one time? What is the probability you hire exactly $n$ times?

You hire once when the best candidate is first. There is $1/n$ chance of that happening ($n!$ positions in $(n-1)!$ of which they are first).

You hire $n$ times when the candidates come in increasing order, which is $1/n!$.