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 9.5 - Page 582: 14

Answer

a. (16 choose 7) = 16!/7!(9!) = 10*11*12*13*14*15*16/(2*3*4*5*6*7) = 10*11*13*8 = 11440 b. (16 choose 13) + (16 choose 14) + (16 choose 15) + (16 choose 16) = 16!/13!3! + 16!/14!2! + 16!/15! + 1 = 14*15*16/2*3 + 15*16/2 + 16 + 1 = 7*5*16 + 15*8 + 16 + 1 = 560 + 120 + 16 + 1 = 697 c. [total bit strings] – [total bit strings that don’t contain 1] = 216 - (16 choose 0) = 216 – 16!/0!(16!) = 216 – 1 = 65535 d. (16 choose 1) + (16 choose 0) = 16!/1!(15!) + 1 = 16 + 1 = 17

Work Step by Step

a. (16 choose 7) = 16!/7!(9!) = 10*11*12*13*14*15*16/(2*3*4*5*6*7) = 10*11*13*8 = 11440 b. (16 choose 13) + (16 choose 14) + (16 choose 15) + (16 choose 16) = 16!/13!3! + 16!/14!2! + 16!/15! + 1 = 14*15*16/2*3 + 15*16/2 + 16 + 1 = 7*5*16 + 15*8 + 16 + 1 = 560 + 120 + 16 + 1 = 697 c. [total bit strings] – [total bit strings that don’t contain 1] = 216 - (16 choose 0) = 216 – 16!/0!(16!) = 216 – 1 = 65535 d. (16 choose 1) + (16 choose 0) = 16!/1!(15!) + 1 = 16 + 1 = 17
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