The Coupon Collector problem

Introduction

Suppose your favorite brand of cereals is having a competition, where in each box of cereals there is a coupon with a positive integer number between 1 and N, and you win a prize if you collect coupons with all numbers between 1 and N. Moreover, when you buy a new box of cereals, the coupon inside the box independently from other boxes has the number i with probability 1/N, for every i = 1, …, N.

The obvious question to ask now is the following.

How many boxes do we need to buy in order to…