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27a-Pigeonhole-Principle

# 27a-Pigeonhole-Principle - Pigeonhole Introduction...

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Pigeonhole Principle Discrete Mathematics

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Discrete Mathematics – Pigeonhole Principle 27-2 Pigeonhole Principle If m pigeons occupy n pigeonholes and m > n, then at least one pigeonhole has two or more pigeons roosting in it.
Discrete Mathematics – Pigeonhole Principle 27-3 Examples Among any group of 367 (or more) people, there must be at least two with the same birthday, because there are only 366 possible birthdays In any group of 27 English words, there must be at least two that begin with the same letter, because there are 26 letter in the Latin alphabet A function f from a set with k + 1 elements to a set with k elements is not one-to-one

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Discrete Mathematics – Pigeonhole Principle 27-4 More Examples Recall that a number c is called the remainder of a when divided by n if -- 0 c < n -- a = k · n + c for some k Note that two numbers, a and b, have the same remainder when divided by n if a – b is divisible by n Among any group of n + 1 integers, there must be at least two with the same remainder when divided by n
Discrete Mathematics – Pigeonhole Principle 27-5 More Examples For every integer n there is a multiple of n that has only zeros and ones in its decimal expansion Solution: Let n be a positive integer. Consider the n + 1 integers 1, 11, 111, … , 11…1 (where the last integer in this list is the integer with n + 1 ones in its decimal expansion). At least two of them have the same remainder when divided by n.

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