4 The Pigeonhole Principle
If \(n + 1\) objects are distributed into \(n\) boxes, then at least one box contains two or more of the objects.
Suppose that no box contains more than one object. Then each box contains at most one object, so the total number of objects is at most \(n\). This contradicts the assumption that there are \(n + 1\) objects.
Let \(q_1, q_2, . .. ,q_n\) be positive integers. If
objects are distributed into \(n\) boxes, then either the first box contains at least \(q_1\) objects, or the second box contains at least \(q_2\) objects, \(\dots \), or the \(n\)th box contains at least \(q_n\) objects.
Let \(n\) and \(r\) be positive integers. If \(n(r-1) + 1\) objects are distributed into \(n\) boxes, then at least one of the boxes contains \(r\) or more of the objects.
If \(m \geq 2\) and \(n \geq 2\) are integers, then there is a positive integer \(p\) such that