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

8.6 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

11.4 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