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Counting & Probability addresses topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and is critical in preparing for prestigious middle and high school math contests such as MATHCOUNTS, AMC 8 and 10, MATHLEAGUE, and AIME.
Chapters and Topics
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
4 Counting and Combinations
4.1 Introduction
4.2 Committee Forming
4.3 How to Compute Combinations
4.4 Our First Combinatorial Identity
4.5 Summary
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
6 Some Harder Counting Problems
6.1 Introduction
6.2 Problems
6.3 Summary
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
9 Think About It!
9.1 Introduction
9.2 Problems
9.3 Summary
Challenge Problems
10 Geometric Probability
10.1 Introduction
10.2 Probability Using Lengths
10.3 Probability Using Areas
10.4 Summary
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
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
Review Problems
Challenge Problems
15 More Challenging Problems
15.1 Introduction
15.2 Problems
Challenge Problems
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