Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 9 - Counting and Probability - Exercise Set 9.4 - Page 563: 3

Answer

Yes. $500>366$, so by pigeonhole principle this is true.

Work Step by Step

There are 366 available birthdays for residents and 500 people. Therefore, by pigeonhole principle, at least one pair must share a birthday. This is because after assigning 1 birthday per person for the 1st 366 in an attempt to avoid sharing birthdays, the next person has no possible birthday, so they must share a birthday with someone. The only way to avoid this is to have someone before share a birthday, so a pair of people must always share a birthday.
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