Introduction to Counting & Probability (The Art of Problem Solving)

✍ Scribed by David Patrick

• Publisher: AoPS Inc
• Year: 2007
• Tongue: English
• Leaves: 256
• Edition: 2
• Category: Library

✦ Synopsis

Book by Patrick, David

✦ Table of Contents

Contents
1 Counting Is Arithmetic
1.1 Introduction
1.2 Counting Lists of Numbers
1.3 Counting with Addition and Subtraction
1.4 Counting Multiple Events
1.5 Permutations
1.6 Summary
2 Basic Counting Techniques
2.1 Introduction
2.2 Casework
2.3 Complementary Counting
2.4 Constructive Counting
2.5 Counting with Restrictions
2.6 Summary
3 Correcting for Overcounting
3.1 Introduction
3.2 Permutations with Repeated Elements
3.3 Counting Pairs of Items
3.4 Counting with Symmetries
3.5 Summary
4 Committees and Combinations
4.1 Introduction
4.2 Committee Forming
4.3 How to Compute Combinations
4.4 Our First Combinatorial Identity
4.5 Summary
5 More With Combinations
5.1 Introduction
5.2 Paths on a Grid
5.3 More Committee-type Problems
5.4 Distinguishability
5.5 Summary
6 Some Harder Counting Problems
6.1 Introduction
6.2 Problems
6.3 Summary
7 Introduction to Probability
7.1 Introduction
7.2 Basic Probability
7.3 Equally Likely Outcomes
7.4 Counting Techniques in Probability Problems
7.5 Summary
8 Basic Probability Techniques
8.1 Introduction
8.2 Probability and Addition
8.3 Complementary Probabilities
8.4 Probability and Multiplication
8.5 Probability with Dependent Events
86 Shooting Stars — a hard problem
8.7 Summary
9 Think About It!
9.1 Introduction
9.2 Problems
9.3 Summary
10 Geometric Probability
10.1 Introduction
10.2 Probability Using Lengths
10.3 Probability Using Areas
10.4 Summary
11 Expected Value
11.1 Introduction
11.2 Definition of Expected Value
11.3 Expected Value Problems
114 A Funky Game
11.5 Summary
12 Pascal’s Triangle
12.1 Introduction
12.2 Constructing Pascal’s Triangle
12.3 Those Numbers Look Familiar!
12.4 An Interesting Combinatorial Identity
12.5 Another Interesting Combinatorial Identity
12.6 Summary
13 The Hockey Stick Identity
13.1 Introduction
13.2 The Problem
13.3 A Step-by-Step Solution
13.4 A Clever Solution
13.5 The Identity
13.6 Summary
14 The Binomial Theorem
14.1 Introduction
14.2 A Little Algebra
14.3 The Theorem
14.4 Applications of the Binomial Theorem
14.5 Using the Binomial Theorem in Identities
14.6 Summary
15 More Challenging Problems
15.1 Introduction
15.2 Problems
Hints to Selected Problems
Index