In combinatorics, the pigeonhole principle states that if or more pigeons are placed into holes, one hole must contain two or more pigeons. This seemingly trivial statement may be used with remarkable creativity to generate striking counting arguments, especially in Olympiad settings.
In older texts, the principle may be referred to as the Dirichlet box principle. A common phrasing of the principle uses balls and boxes and is that if balls are to be placed in boxes and , then at least one box must contain more than one ball.
The pigeonhole principle may be demonstrated intuitively by the following argument: assume that there exists a way to place balls into boxes, where such that all boxes contain at most one ball.
Let how many balls each box contains. Our condition that all boxes contain at most one ball implies that for all , so However, we know that there are a total of balls across all our boxes, so this sum must equal : Therefore, . This contradicts our condition that . Therefore, our assumption is incorrect; at least one box must contain two or more balls.
In formal terms, the pigeonhole principle is a consequence of how one set is defined to be larger than another set.
Let be a set of balls and be a set of boxes such that (or equivalently, ). The definition of this is that there exists a surjective mapping from to , but not an injection. In other words, there exists a way to map every ball of to every box of , but it does not hold that if the boxes of two balls are the same, then the balls must be the same. That is to say, there must be two or more balls in the same box, which is the pigeonhole principle.
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