Pigeonhole principle
Hand shaking

If n people can shake hands with one another (where n > 1), the pigeonhole principle shows that there is always a pair of people who will shake hands with the same number of people. In this application of the principle, the "hole" to which a person is assigned is the number of hands that person shakes. Since each person shakes hands with some number of people from 0 to n − 1, there are n possible holes. On the other hand, either the "0" hole, the " n − 1" hole, or both must be empty, for it is impossible (if n > 1) for some person to shake hands with everybody else while some person shakes hands with nobody. This leaves n people to be placed into at most n − 1 non-empty holes, so the principle applies.

This hand-shaking example is equivalent to the statement that in any graph with more than one vertex, there is at least one pair of vertices that share the same degree. This can be seen by associating each person with a vertex and each edge with a handshake.
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