# Combinatorics: An Introduction

## Books 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.

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

1.6 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

11.1 Venn Diagrams and Permutations 111

11.2 Exercises 120

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

14.1 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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