Combinatorics : an introduction

**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. Cover 1 Title Page 5 Copyright 6 Contents 9 Preface 13 Chapter 1: Logic 17 1.1 Formal Logic 17 1.2 Basic Logical Strategies 23 1.3 The Direct Argument 27 1.4 More Argument Forms 29 Converse Statements 29 Contrapositive Statements 30 Counterexamples 32 1.5 Proof by Contradiction 33 1.6 Exercises 42 Chapter 2: Sets 43 2.1 Set Notation 43 2.2 Predicates 44 2.3 Subsets 46 2.4 Union and Intersection 48 2.5 Exercises 51 Chapter 3: Venn Diagrams 53 3.1 Inclusion-Exclusion Principle 53 3.2 Two-Circle Venn Diagrams 56 3.3 Three-Square Venn Diagrams 60 3.4 Exercises 68 Chapter 4: Multiplication Principle 73 4.1 What Is the Principle? 73 4.2 Exercises 79 Chapter 5: Permutations 81 5.1 Some Special Numbers 82 5.2 Permutations Problems 83 5.3 Exercises 87 Chapter 6: Combinations 89 6.1 Some Special Numbers 89 6.2 Combination Problems 90 6.3 Exercises 95 Chapter 7: Problems Combining Techniques 97 7.1 Significant Order 97 7.2 Order Not Significant 98 7.3 Exercises 105 Chapter 8: Arrangement Problems 107 8.1 Examples of Arrangements 108 8.2 Exercises 114 Chapter 9: At Least, At Most, and Or 115 9.1 Counting with Or 115 9.2 At Least, At Most 121 9.3 Exercises 125 Chapter 10: Complement Counting 127 10.1 The Complement Formula 127 10.2 A New View of "At Least" 130 10.3 Exercises 134 Chapter 11: Advanced Permutations 137 11.1 Venn Diagrams and Permutations 137 11.2 Exercises 152 Chapter 12: Advanced Combinations 155 12.1 Venn Diagrams and Combinations 155 12.2 Exercises 164 Chapter 13: Poker and Counting 165 13.1 Warm-Up Problems 166 13.2 Poker Hands 168 13.3 Jacks or Better 175 13.4 Exercises 177 Chapter 14: Advanced Counting 179 14.1 Indistinguishable Objects 179 14.2 Circular Permutations 183 14.3 Bracelets 186 14.4 Exercises 191 Chapter 15: Algebra and Counting 193 15.1 The Binomial Theorem 193 15.2 Identities 196 15.3 Exercises 202 Chapter 16: Derangements 203 16.1 Mathematical Induction 204 16.2 Fixed-Point Theorems 208 16.3 His Own Coat 213 16.4 Inclusion/Exclusion for Many Sets 214 16.5 A Common Miscount 219 16.6 Exercises 222 Chapter 17: Probability Vocabulary 223 17.1 Vocabulary 223 Chapter 18: Equally Likely Outcomes 229 18.1 Outcomes in Experiments 229 18.2 Exercises 236 Chapter 19: Probability Trees 239 19.1 Tree Diagrams 239 19.2 Exercises 248 Chapter 20: Independent Events 251 20.1 Independence 251 20.2 Logical Consequences of Influence 254 20.3 Exercises 258 Chapter 21: Sequences and Probability 261 21.1 Sequences of Events 261 21.2 Exercises 267 Chapter 22: Conditional Probability 269 22.1 What Does Conditional Mean? 269 22.2 Exercises 274 Chapter 23: Bayes' Theorem 277 23.1 The Theorem 277 23.2 Exercises 282 Chapter 24: Statistics 285 24.1 Introduction 285 24.2 Probability Is Not Statistics 285 24.3 Conversational Probability 286 24.4 Conditional Statistics 294 24.5 The Mean 297 24.6 Median 298 24.7 Randomness 300 Chapter 25: Linear Programming 307 25.1 Continuous Variables 308 25.2 Discrete Variables 312 25.3 Incorrectly Applied Rules 316 Chapter 26: Subjective Truth 319 26.1 The Absolute Truth of an Axiom 319 Bibliography 325 Index 327 Machine generated contents note: 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 . 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. The EPUB format of this title may not be compatible for use on all handheld devices. "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"-- "This book provides a treatment of counting combinatorics and contains topical discussions beyond what is typically seen in other related books. Formulas are discussed and justified, and examples include unique approaches and ideas to the discussed topics"-- "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"-- Provided by publisher