4_the_Pigeonhole_Principle_the_Binomial_Theorem

4_the_Pigeonhole_Principle_the_Binomial_Theorem - Discrete...

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Unformatted text preview: Discrete Mathematics Dr. Fred Phelps Lecture 4 The Pigeonhole Principle & The Binomial Theorem LVP 2.4 Pigeonhole Principle • If we have n boxes and we place more than n objects into them, then there will be at least one box that contains more than one object. • Pigeon holes are wooden boxes used for mail in offices. • In Russian and some other languages, it is known as the "Dirichlet principle“. LPV 2.4 Application of the Pigeonhole Principle – Number of Hair Strands Problem: Prove that at least two people in New York City have the same number of strands of hair on their heads. • Nobody knows exactly how many strands of hair ( N ) are on their own head. • But biologists tell us that N <500,000. • The population of New York City (NYC) is over ten million. • There must be some value of N for which Claim: In a group of at least two people, there are always at least two who shake the same number of people’s hands. • Let the k th box B( k ) be the number of people shaking k hands, k = 0, 1, … n-1. So there are n boxes. • Note that • So one of the n boxes, either • At most n-1 “boxes” are “not empty”. if (0) 0 then ( -1) 0. B B n ≠ = ( 29 ( 29 0 or 1 0. B B n =-= Application of the Pigeonhole Principle - Hand Shaking LPV 2.4 Application of the Pigeonhole Principle – Distance between Shots Problem: Given that 50 shots hit a 70x70 cm target what can we say about how close two of them must be?...
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This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Spring '11 term at Jordan University of Science & Tech.

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4_the_Pigeonhole_Principle_the_Binomial_Theorem - Discrete...

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