This book provides a treatment of counting combinatorics that uniquely includes detailed formulas, proofs, and exercises and features coverage of derangements, elementary probability, conditional probability, independent probability, and Bayes Theorem. Using elementary applications that never advance beyond the use of Venn diagrams, the inclusion/exclusion formula, the multiplication principal, permutations, and combinations, Combinatorics is perfect for courses on discrete or finite mathematics-or as a reference for anyone who wants to learn about the various applications of elementary combinatorics.

**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.

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