Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 6 - Section 6.2 - The Pigeonhole Principle - Exercises - Page 405: 14

Answer

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Work Step by Step

$\underline{Pigeonhole}$ $ \underline{Principle}$- If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects. a) Consider the following tuples (1,10), (2,9), (3,8), (4,7), (5,6). Look at the first entries of the tuples. These are five integers. If we choose 6 integers, then by pigeon hole principle, there is at least one pair that add up to 11. If we choose 5 integers, we may not get a pair that adds up to give 11 say {1,2,3,4,5}. If we choose 7 integers, then by pigeon hole principle there are 2 pairs that add up to 11. b) No this is not true. look at the set {1,2,3,4,5,6}. This contains only one pair that adds up to 11.
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