Introduction to Counting & Probability (The Art of Problem Solving) [2 ed.] 1934124109, 9781934124109

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

Citation preview

CONTENTS

HIIIIREREIREEIEHR Contents

How to Use This Book

iii

Acknowledgements

vii

1

Counting Is Arithmetic 1.1 1.2 1.3 1.4 1.5 1.6

2

Basic Counting Techniques 2.1 2.2 2.3 2.4 2.5 2.6

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Lists of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting with Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Multiple Events Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Casework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complementary Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructive Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting with Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Correcting for Overcounting 3.1 3.2 3.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutations with Repeated Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Pairs of Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 6 13 19 21

27 27 27 35 38 41 45

49 49 49 53 ix

CONTENTS

Counting with Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 3.5

4

Committees and Combinations Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Committee Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to Compute Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Our First Combinatorial Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 4.2 4.3 4.4 4.5

5

More With Combinations 5.1 5.2 5.3 5.4 5.5

6

Some Harder Counting Problems

Introduction to Probability 7.1 7.2 7.3 7.4 7.5

8

65 65 65 69 72 76

81 81 81 84 88 92

97

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1 6.2 6.3

7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paths on a Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Committee-type Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 62

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equally Likely Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Techniques in Probability Problems . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Basic Probability Techniques 8.1 8.2 8.3

111 111 112 115 118 120

123

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Probability and Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Complementary Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

CONTENTS

8.4 8.5 86* 8.7

Probability and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability with Dependent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shooting Stars — a hard problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Think About It!

9

9.1 9.2 9.3

1 0 10.1 10.2 10.3 10.4

1 1 11.1 11.2 11.3 114* 11.5

1 2 12.1 12.2 12.3 12.4 12.5 12.6

1 3 13.1

130 136 138 141

145

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Geometric Probability

153

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability Using Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability Using Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 154 156 162

Expected Value

165

Introduction . . . . . . . . . Definition of Expected Value Expected Value Problems . . A Funky Game . . . . . . . . Summary . . . . . . . . . . .

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175

Pascal’s Triangle Introduction . . . . . . . . . . . . . . . . . . Constructing Pascal’s Triangle . . . . . . . . Those Numbers Look Familiar! . . . . . . . An Interesting Combinatorial Identity . . . Another Interesting Combinatorial Identity Summary . . . . . . . . . . . . . . . . . . . .

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. 175 . 176 . 177 . 180 . 184 . 187

The Hockey Stick Identity Introduction

165 166 166 169 171

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191 191

CONTENTS

13.2 13.3 13.4 13.5 13.6

1 4 14.1 14.2 14.3 14.4 14.5 14.6

1 5 15.1 15.2

The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Step-by-Step Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Clever Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192 193 198 199 205

The Binomial Theorem

209

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Little Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of the Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using the Binomial Theorem in Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 209 211 213 214 216

More Challenging Problems

219

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Hints to Selected Problems

231

Index

241

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