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 406: 41

Answer

The wrestler played exactly 24 matches between consecutive hours.

Work Step by Step

$x_{i}$ = number of matches after $i$-th hour We know total number of hours = 75 So 1$\leq$i$\leq$75 and 1$\leq$$x_{i}$$\leq$125. Adding 24 to $x_{i}$ we get 25$\leq$$x_{i}$+24$\leq$149 Thus $x_{1}$,$x_{2}$...$x_{75}$,$x_{1}$+24,$x_{2}$+24..$x_{75}$+24(150integers) are between 1 and 149. by pigeonhole principle $x_{i}$+24 = $x_{j}$where i$\ne$j Hence proved
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