Consider \(n\) items \(B_ 1, B_ 2, \dots, B_ N\), and suppose that at each step one of them is chosen and removed to the left-hand end, while other items move to the right. If the choices are independent, then the successive configurations of the items form a Markov chain, known as the single shelf library chain. The author derives the \(n\)-step probabilities of the chain with a new method based on the close connection of the described scheme to the coupon-collector’s problem.