Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 11 - Counting Methods and Probability Theory - 11.3 Combinations - Exercise Set 11.3: 39

Answer

$4,516,932,420$ committees

Work Step by Step

Committee members are chosen with no importance of order of choice, so we deal with combinations. We have a sequence of selections in which we choose 1. ... 4 out of a group of 55 Republicans ... in ${}_{55}C_{4}$ ways 2. ... 3 out of a group of 44 Democrats... in ${}_{44}C_{3}$ ways By the Fundamental Counting Principle, Total ways= ${}_{55}C_{4}\cdot {}_{44}C_{3}$ ${}_{55}C_{4}=\displaystyle \frac{55!}{(55-4)!4!}$ $=\displaystyle \frac{55\times 54\times 53\times 52}{1\times 2\times 3\times 4}=341,055$ ${}_{44}C_{3}=\displaystyle \frac{44!}{(44-3)!3!}=\frac{44\times 43\times 42}{1\times 2\times 3}=13,244$ Total=$341,055\times 13,244$= $4,516,932,420$
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