Here is the beginning of a tree diagram that may help illustrate the menu options.

 

The pigeonhole principle relies on filling existing spaces (pigeonholes, boxes, envelopes, and the like) with items (pigeons, coins, letters, and so on) to the point where all spaces are just one item short of being full. At this point, no matter where the next item is placed, we get a full box. The variables here are the number of spaces to fill and the number of items required in each space to assure that space is full.

A generalization of the situations we have considered is the following:

Let n represent the number of spaces to fill and use to represent the number of items needed for each of the n spaces to be considered full. Then to reach the point where each of the n spaces are just one item short of being full, we need this many items:

The next item we add, no matter where we put it, will create a full space. Therefore, the number of items needed to assure that at least one of the n spaces is full is.