Inclusion Exclusion Principle Combinatorics

Question

My problem is: Assume that every time you buy a box of cereal, you receive one of the pictures of the New York Yankees players and there are $n$ on the team. Over some time you buy $m\ge$$n$ boxes of cereal. Use the Inclusion-Exclusion principle to show that the probability that you will get all $n$ pictures is: $1- {n \choose 1}({n-1 \over n})^m + {n \choose 2}({n-2 \over n})^m \ldots + (-1)^{n-1}{n \choose n-1}({1 \over n})^m $

I am new to this principle and to combinatorics in general so I may be missing some key information or obvious details. I understand that we have a point of reference so that the first term will always be $1$ but I do not understand how we can generate the other terms or where they come from in context of the problem.

Answer 1

Hint

For $i = 1, 2, \ldots, n$, let $E_i$ be the event that you do not have a picture of player $i$. By inclusion–exclusion, $$P(E_1 \cup \cdots \cup E_n) = \sum_{1 \leq i \leq n} P(E_i) - \sum_{1 \leq i < j \leq n} P(E_i \cap E_j) + \sum_{1 \leq i < j < k \leq n} P(E_i \cap E_j \cap E_k) - \cdots.$$ Now note that $E_1 \cap E_2 \cap \cdots \cap E_r$ is the event that $m$ independently chosen cereal boxes all yielded a photograph from the same set of $n-r$ players. What is the chance of this? Would it make a difference if the set of players were different, but still the same size?