Editing Pigeonhole principle (section)

=== Hand shaking ===
If {{math|''n''}} people can shake hands with one another (where {{math|''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 {{math|''n'' &minus; 1}}, there are {{math|''n''}} possible holes. On the other hand, either the "0" hole, the {{math|"''n'' &minus; 1"}} hole, or both must be empty, for it is impossible (if {{math|''n'' > 1}}) for some person to shake hands with everybody else while some person shakes hands with nobody. This leaves {{math|''n''}} people to be placed into at most {{math|''n'' &minus; 1}} non-empty holes, so the principle applies.

This hand-shaking example is equivalent to the statement that in any [[Graph (discrete mathematics)|graph]] with more than one [[Vertex (graph theory)|vertex]], there is at least one pair of vertices that share the same [[Degree (graph theory)|degree]].<ref>{{Cite web|last=Pandey|first=Avinash|title=D3 Graph Theory - Interactive Graph Theory Tutorials|url=https://d3gt.com/unit.html?pigeonhole|access-date=2021-01-12|website=d3gt.com|language=en}}</ref> This can be seen by associating each person with a vertex and each [[Edge (graph)|edge]] with a handshake.{{Citation needed|date=February 2025}}