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Combinatorics: An Introduction

ISBN: 978-1-118-40436-2
328 pages
January 2013
Combinatorics: An Introduction (111840436X) cover image

Bridges combinatorics and probability and uniquely includes detailed formulas and proofs to promote mathematical thinking

Combinatorics: An Introduction introduces readers to counting combinatorics, offers examples that feature unique approaches and ideas, and presents case-by-case methods for solving problems.

Detailing how combinatorial problems arise in many areas of pure mathematics, most notably in algebra, probability theory, topology, and geometry, this book provides discussion on logic and paradoxes; sets and set notations; power sets and their cardinality; Venn diagrams; the multiplication principal; and permutations, combinations, and problems combining the multiplication principal. Additional features of this enlightening introduction include:

  • Worked examples, proofs, and exercises in every chapter
  • Detailed explanations of formulas to promote fundamental understanding
  • Promotion of mathematical thinking by examining presented ideas and seeing proofs before reaching conclusions
  • Elementary applications that do not advance beyond the use of Venn diagrams, the inclusion/exclusion formula, the multiplication principal, permutations, and combinations

Combinatorics: An Introduction is an excellent book for discrete and finite mathematics courses at the upper-undergraduate level. This book is also ideal for readers who wish to better understand the various applications of elementary combinatorics.

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Preface xiii

1 Logic 1

1.1 Formal Logic  1

1.2 Basic Logical Strategies  6

1.3 The Direct Argument  10

1.4 More Argument Forms  12

1.5 Proof By Contradiction  15

1.6 Exercises   23

2 Sets 25

2.1 Set Notation  25

2.2 Predicates   26

2.3 Subsets   28

2.4 Union and Intersection  30

2.5 Exercises   32

3 Venn Diagrams 35

3.1 Inclusion/Exclusion Principle  35

3.2 Two Circle Venn Diagrams  37

3.3 Three Square Venn Diagrams  42

3.4 Exercises   50

4 Multiplication Principle 55

4.1 What is the Principle?  55

4.2 Exercises   60

5 Permutations 63

5.1 Some Special Numbers  64

5.2 Permutations Problems  65

5.3 Exercises   68

6 Combinations 69

6.1 Some Special Numbers  69

6.2 Combination Problems  70

6.3 Exercises   74

7 Problems Combining Techniques 77

7.1 Significant Order  77

7.2 Order Not Significant  78

7.3 Exercises   83

8 Arrangement Problems 85

8.1 Examples of Arrangements  86

8.2 Exercises   91

9 At Least, At Most, and Or 93

9.1 Counting With Or  93

9.2 At Least, At Most  98

9.3 Exercises   102

10 Complement Counting 103

10.1 The Complement Formula  103

10.2 A New View of ?At Least?  105

10.3 Exercises   109

11 Advanced Permutations 111

11.1 Venn Diagrams and Permutations  111

11.2 Exercises   120

12 Advanced Combinations 125

12.1 Venn Diagrams and Combinations  125

12.2 Exercises   131

13 Poker and Counting 133

13.1 Warm Up Problems  133

13.2 Poker Hands   135

13.3 Jacks or Better  141

13.4 Exercises   143

14 Advanced Counting 145

14.1 Indistinguishable Objects  145

14.2 Circular Permutations  148

14.3 Bracelets   151

14.4 Exercises   155

15 Algebra and Counting 157

15.1 The Binomial Theorem  157

15.2 Identities   160

15.3 Exercises   165

16 Derangements 167

16.1 Fixed Point Theorems  168

16.2 His Own Coat  173

16.3 Exercises   174

17 Probability Vocabulary 175

17.1 Vocabulary   175

18 Equally Likely Outcomes 181

18.1 Exercises   188

19 Probability Trees 189

19.1 Tree Diagrams  189

19.2 Exercises   198

20 Independent Events 199

20.1 Independence  199

20.2 Logical Consequences of Influence  202

20.3 Exercises   206

21 Sequences and Probability 209

21.1 Sequences of Events  209

21.2 Exercises   215

22 Conditional Probability 217

22.1 What Does Conditional Mean?  217

22.2 Exercises   223

23 Bayes? Theorem 225

23.1 The Theorem  225

23.2 Exercises   230

24 Statistics 231

24.1 Introduction   231

24.2 Probability is not Statistics  231

24.3 Conversational Probability  232

24.4 Conditional Statistics  239

24.5 The Mean   241

24.6 Median   242

24.7 Randomness   244

25 Linear Programming 249

25.1 Continuous Variables  249

25.2 Discrete Variables  254

25.3 Incorrectly Applied Rules  258

26 Subjective Truth 261

Bibliography 267

Index 269

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THEODORE G. FATICONI, PhD, is Professor in the Department of Mathematics at Fordham University. His professional experience includes forty research papers in peer-reviewed journals and forty lectures on his research to colleagues.

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