Learn more about the Pigeonhole Principle and other powerful techniques for combinatorics problems in our Intermediate Counting & Probability textbook by USA Math Olympiad winner (and MIT PhD) David Patrick.
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Pigeonhole Principle

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Pigeonhole Principle

The basic pigeonhole principle says that if there are $n$ holes, and $n+k$ piegons (k>1), then one hole MUST contain two or more pigeons. The extended version of the pigeonhole principle states that for n holes, and $nk+j$ pigeons, j>1, some hole must contain k+1 pigeons. If you see a problem with the numbers n, and nk+1, think about pigeonhole.