Table Of ContentC O U N T I N G
Solutions Manual 2nd Edition
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C O U N T I N G
Solutions Manual 2nd Edition
Koh Khee Meng
National University of Singapore, Singapore
Tay Eng Guan
Nanyang Technological University, Singapore
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Koh, K. M. (Khee Meng), 1944–
Counting : supplementary notes and solutions manual / Koh Khee Meng, Tay Eng Guan.
p. cm.
Includes bibliographical references.
ISBN 981-256-915-4 (pbk: alk. paper)
1. Combinatorial analysis. 2. Counting. 3. Graph theory. I. Title. II. Tay, Eng Guan,
QA164 .K63 2002 Suppl.
511'.6--dc22 2006049623
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Counting: Solutions Manual (2nd Edition)
ISBN: 978-981-4401-94-4 (pbk)
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
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Typeset by Stallion Press
Email: [email protected]
Printed in Singapore.
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Preface
Combinatorics is a branch of mathematics dealing with discretely
structured problems. Its scope of study includes selections and
arrangements of objects with prescribed conditions, configurations
involving a set of nodes interconnected by edges (called graphs),
and designs of experimental schemes according to specified rules.
Combinatorialproblemsandtheirapplicationscanbefoundnotonly
invariousbranchesofmathematics, butalsoinotherdisciplinessuch
as engineering, computer science, operational research, management
sciencesandthelifesciences.Sincecomputersrequirediscreteformu-
lation of problems, combinatorial techniques have become essential
and powerful tools for engineers and applied scientists, in particu-
lar, in the area of designing and analysing algorithms for various
problems which range from designing the itineraries for a shipping
company to sequencing the human genome in the life sciences.
Thecounting problem, whichseekstofindouthowmanyarrange-
ments thereare in aparticular situation, is oneof thebasic problems
incombinatorics. Countingisusedinforensicsciencetocalculate the
probability that a sample of biological evidence found at the crime
scene matches that of a particular accused person. In Chemistry,
Cayley used graphs to count the number of isomers of saturated
hydrocarbons, while Harary and Read counted the number of cer-
tain organic compounds built up from benzene rings by representing
them as configurations of hexagons joined together along a common
edge. In Genetics, bycounting all possibilities for aDNA chain made
upofthefourbases,scientistsarriveatanastoundinglylargenumber
and so are able to understand the tremendous possible variation in
geneticmakeup.Countinghasbeenusedaswelltostudytheprimary
and secondary structures of RNA.
v
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vi Counting: Solutions Manual
This book is the essential companion to Counting (2nd Edi-
tion) (World Scientific, 2013), an introduction to combinatorics for
secondarytoundergraduatestudents.Thebookgivessolutionstothe
exercises in Counting (2nd Edition). There is often more than one
method to solve a particular problem and the authors have included
alternative solutions whenever they are of interest.
Problems marked with (AIME) are from the American Invita-
tional Mathematics Examination. We would like to express our grat-
itude to Mathematical Association of America for allowing us to
include these problems in the book. Many thanks go to our col-
leagues, Dong Fengming, Ku Cheng Yeaw, Lee Tuo Yeong, Ng Kah
Loon, Toh Pee Choon, Toh Tin Lam and Katherine Goh for reading
throughthedraftandcheckingthroughthesolutions—anymistakes
that remain are ours alone.
Koh Khee Meng
Tay Eng Guan
September 2012
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Contents
Preface v
Notation and Abbreviation ix
Po´lya’s Model of Problem Solving 1
1. The Addition Principle 7
2. The Multiplication Principle 15
3. Subsets and Arrangements 23
4. Applications 29
5. The Bijection Principle 41
6. Distribution of Balls into Boxes 51
7. More Applications of (BP) 55
8. Distribution of Distinct Objects into Distinct Boxes 65
9. Other Variations of the Distribution Problem 69
10. The Binomial Expansion 75
11. Some Useful Identities 81
12. Pascal’s Triangle 85
13. The Principle of Inclusion and Exclusion 91
14. General Statement of the Principle of Inclusion
and Exclusion 97
vii
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viii Counting: Solutions Manual
15. The Pigeonhole Principle 107
16. Recurrence Relations 115
17. The Stirling Numbers of the First Kind 123
18. The Stirling Numbers of the Second Kind 129
19. The Catalan Numbers 133
20. Solutions to Miscellaneous Problems 141
Recommended Books for Further Reading 213
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Notation and Abbreviation
N = {1,2,3,...}
Nk = {1,2,3,...,k}
(AP) : Addition Principle
(MP) : Multiplication Principle
(CP) : Complementation Principle
(IP) : Injection Principle
(BP) : Bijection Principle
(PP) : Pigeonhole Principle
(GPP) : Generalised Pigeonhole Principle
(PIE) : Principle of Inclusion and Exclusion
(GPIE) : Generalised Principle of Inclusion and Exclusion
(BT) : Binomial Theorem
LHS : Left-hand side
RHS : Right-hand side
|S| = the number of elements in the set S
(cid:1)x(cid:2) = the largest integer less than or equal to x
(cid:3)x(cid:4) = the smallest integer greater than or equal to x
FTA : Fundamental Theorem of Arithmetic
gcd : greatest common divisor
Bn = Bell number
= the number of ways of dividing n distinct objects into
(nonempty) groups
C(n) = Catalan number
= the number of shortest routes from O(0,0) to A(n,n)
which do not cross the diagonal y =x in the
rectangular coordinate system
Dn = the number of derangements of Nn
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x Counting: Solutions Manual
s∗(m, k) = the number of ways of arranging m distinct objects
around k identical circles such that each circle has
at least one object
s(m,k) = Stirling number of the first kind
= the coefficient of xk in the expansion of x(x−1)
(x−2)···(x−(m−1))
S(m,k) = Stirling number of the second kind
= the number of ways of distributing m distinct objects
(cid:1) (cid:2) into k identical boxes such that no box is empty
Cn = n = the number of r-element subsets of an n-element set
r r
Pn = the number of r-permutations of n distinct objects
r
AIME : American Invitational Mathematics Examination
