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