The coupon collector's problem

n this note we will consider the following problem: how many coupons we have to purchase (on average) to complete a collection. This problem, which takes every- body back to his childhood when this was really “a prob- lem”, has been considered by the probabilists since the eighteenth century and nowadays it is still possible to de- rive some new results, probably original or at least never published. We will present some classic results, some new formulas, some alternative approaches to obtain known re- sults and a couple of amazing expressions.