Patrick D. Intermediate Counting and Probability

AoPS Incorporated, 2018. — 386 p. — ISBN 9781934124062, 1934124060Learn the basics of counting and probability from former USA Mathematical Olympiad winner David Patrick. Topics covered in the book include permutations, combinations, Pascal's Triangle, basic combinatorial identities, expected value, fundamentals of probability, geometric probability, the Binomial Theorem, and much more.
As you'll see in the excerpts below, the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which counting and probability techniques are taught. Important facts and powerful problem solving approaches are highlighted through out the text. In addition to the instructional material, the book contains over 400 problems. The solutions manual contains full solutions to all of the problems, not just answers.
This book is ideal for students who have mastered basic algebra, such as solving linear equations. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of counting and probability will find this book an instrumental part of their mathematics libraries.
AoPS 2-Book Set : Art of Problem Solving AoPS Intermediate Counting and Probability Textbook and Solutions Manual 2-Book Set : Continue your exploration of more advanced counting and probability topics from former USA Mathematical Olympiad winner David Patrick. This book is the follow-up to the acclaimed Introduction to Counting & Probability textbook. Topics covered in this book include inclusion-exclusion, 1-1 correspondences, the Pigeonhole Principle, constructive expectation, Fibonacci and Catalan numbers, recursion, conditional probability, generating functions, graph theory, and much more. As with all of the books in Art of Problem Solving's Introduction and Intermediate series, the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which counting and probability techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text.How to Use This Book
Acknowledgements
Review of Counting & Probability BasicsBasic Counting Techniques
Basic Probability Techniques
Expected Value
Pascal's Triangle and the Binomial Theorem
Summation NotationSets and LogicSets
Operations on Sets
Truth and Logic
QuantifiersA Piece of PIEPIE With 2 Properties
PIE With 3 Properties
Counting Problems With PIE
PIE With Many Properties
Counting Items With More Than I of Something
Some Harder PIE ProblemsConstructive Counting and CorrespondencesSome Basic Problems
Harder Constructive Counting Problems
1-1 Correspondence Basics
More Complicated 1-1 Correspondences
Clever 1-1 CorrespondencesThe Pigeonhole PrincipleIt's Just Common Sense!
Basic Pigeonhole Problems
More Advanced Pigeonhole ProblemsConstructive ExpectationBasic Examples
Summing Expectations Constructively
Coat With Many Patches (Reprise)DistributionsBasic Distributions
Distributions With Extra Conditions
More Complicated Distribution ProblemsMathematical Induction
Fibonacci NumbersA Motivating Problem
Some Fibonacci Problems
A Formula for the Fibonacci NumbersRecursionExamples of Recursions
Linear Recurrences
A Hard Recursion Problem
Problems Involving Catalan Numbers
Formulas for the Catalan NumbersConditional ProbabilityBasic Examples of Conditional Probability
Some Definitions and Notation
Harder Examples
Let's Make a Deal!Combinatorial IdentitiesBasic Identities
More IdentitiesEvents With StatesState Diagrams and Random Walks
Events With Infinite States
Two-player Strategy GamesGenerating FunctionsBasic Examples of Generating Functions
The Binomial Theorem (as a Generating Function)
Distributions (as Generating Functions)
The Generating Function for Partitions
The Generating Function for the Fibonacci NumbersGraph TheoryDefinitions
Basic Properties of Graphs
Cycles and Paths
Planar Graphs
Eulerian and Hamiltonian PathsChallenge Problems