Introduction to Counting & Probability

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