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AoPS Introduction Series
Introduction to Counting & Probability
Learn the basics of counting and probability from former USA Mathematical Olympiad winner David Patrick. Topics covered in the book include permutations, combinations, Pascal's Triangle, basic combinatorial identities, expected value, fundamentals of probability, geometric probability, the Binomial Theorem, and much more.
As you'll see in the excerpts below, the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which counting and probability techniques are taught. Important facts and powerful problem solving approaches are highlighted through out the text. In addition to the instructional material, the book contains over 400 problems. The solutions manual contains full solutions to all of the problems, not just answers.
This book is ideal for students who have mastered basic algebra, such as solving linear equations. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of counting and probability will find this book an instrumental part of their mathematics libraries.
As you'll see in the excerpts below, the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which counting and probability techniques are taught. Important facts and powerful problem solving approaches are highlighted through out the text. In addition to the instructional material, the book contains over 400 problems. The solutions manual contains full solutions to all of the problems, not just answers.
This book is ideal for students who have mastered basic algebra, such as solving linear equations. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of counting and probability will find this book an instrumental part of their mathematics libraries.
243 pages, Paperback
First published January 1, 2005
24 people are currently reading
470 people want to read
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Displaying 1 - 11 of 11 reviews
May 11, 2017
Throughout my 2 years of high school, math has never been challenging enough. In order for a challenge, I either had to ask for extra work or, look to outside sources. On top of not being challenged, I also wanted to learn probability and other math subjects, but the school has a set curriculum. When I was searching online I came across the Art of problem solving. For my most recent book club, in English funny enough, I bought the “Intro to counting and probability” book by David Patrick. As you can tell from the book title in the book you learn about probability and counting. Oh yeah, the counting isn't as easy as you would think.
In the first few chapters, it is all about counting. In chapter one, you learn different types of counting like counting lists of numbers, counting with addition and subtraction, counting multiple events and permutation. In just the very first chapter the book already teaches 4 new instances of counting, yes I learned that from the book too. The next chapter is about counting techniques. My favorite techniques that was taught was counting with restrictions. These problems take multiple steps to solve and most of them are solved using factorials of numbers. One of the problems was “A senate committee has 5 republicans and 4 Democrats. In how many ways can the committee members sit in a row of 9 if all 4 democrats sit together.”(David Patrick). To start with this problem without a restriction would be as easy as 9!, however, there is a restriction so it isn't’ that easy. First, we would figure out how many ways there are to order the 4 democrats which would be 4! ways or 4x3x2x1, and the 5! for the 5 republicans. After we know this information, we know that we can either sit the 4 democrats can sit in a row of four 6 ways, so we would multiply the answer by six. The answer would then be 4!x5!x6=17,280 ways to order them, and this is just one way to solve this problem. This problem is one of many they give you in the book, and the examples really help you understand the concepts being taught. And this is just the first 2 chapters!
Overall, this 243 page math book has taught me more math than my 900 page math books in school do. After reading the math book, I didn't just get better at counting and probability, but I got better at math in general. The ways the book teaches not only teaches the topic, but ways and strategies to attack problems and solve them. I would recommend this book to anyone who loves math or is looking to get better at math. I love reading this book and believe you will too!
In the first few chapters, it is all about counting. In chapter one, you learn different types of counting like counting lists of numbers, counting with addition and subtraction, counting multiple events and permutation. In just the very first chapter the book already teaches 4 new instances of counting, yes I learned that from the book too. The next chapter is about counting techniques. My favorite techniques that was taught was counting with restrictions. These problems take multiple steps to solve and most of them are solved using factorials of numbers. One of the problems was “A senate committee has 5 republicans and 4 Democrats. In how many ways can the committee members sit in a row of 9 if all 4 democrats sit together.”(David Patrick). To start with this problem without a restriction would be as easy as 9!, however, there is a restriction so it isn't’ that easy. First, we would figure out how many ways there are to order the 4 democrats which would be 4! ways or 4x3x2x1, and the 5! for the 5 republicans. After we know this information, we know that we can either sit the 4 democrats can sit in a row of four 6 ways, so we would multiply the answer by six. The answer would then be 4!x5!x6=17,280 ways to order them, and this is just one way to solve this problem. This problem is one of many they give you in the book, and the examples really help you understand the concepts being taught. And this is just the first 2 chapters!
Overall, this 243 page math book has taught me more math than my 900 page math books in school do. After reading the math book, I didn't just get better at counting and probability, but I got better at math in general. The ways the book teaches not only teaches the topic, but ways and strategies to attack problems and solve them. I would recommend this book to anyone who loves math or is looking to get better at math. I love reading this book and believe you will too!
December 28, 2018
my big bro gave me this one, too. it's also amazing as well.
May 12, 2017
We’ve all been through a boring math class--mindless memorization of formulas, dull repetition of drills. The company Art of Problem Solving produces textbooks that hopes to counteract any damage these sub-par classes have done. These textbooks challenge even the best of the best students. Instead of routine problems that match each other, the book presents problems that force creative problem solving. In this way, students are continuously challenged to struggle and persevere through problems. The author claims that even if the reader doesn’t go into math, the ability to fight through problems is essential to any career.
My book focused Counting and Probability, an area of math not covered in the normal math curriculum. Even though counting doesn’t sound too difficult, it requires extreme intuition for example, the questions look like “On the island of Mumble, the Mumblian alphabet has only 5 letters, and every word in the Mumblian language has no more than 3 letters in it. How many words are possible?” and “An unfair coin has a 2/3 probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even?” Obviously, these questions require creativity and advanced problem solving. Just from being exposed to these types of problems, I have vastly increased my mathematical capabilities. I recommend this to anyone hoping to improve their creativity and complex thinking.
May 26, 2021
Finally, I've finished this book today! What an adventure it has been! This book has taught me more than a thousand tedious math classes will ever do. Anyone who wants to improve their combinatorics fundamentals should pick up this book.
It not only covers a wide range of topics but also teaches how to approach and solve problems in a variety of ways. This book is recommended for anyone who enjoys math or would like to improve their math skills. It'll be well worth your time.
It not only covers a wide range of topics but also teaches how to approach and solve problems in a variety of ways. This book is recommended for anyone who enjoys math or would like to improve their math skills. It'll be well worth your time.
January 25, 2022
Really great approach with stating the problems before the material, makes it much more interesting to read a chapter to check if your solution was right or not. Also gives a great conceptual way to think about pascals triangle, usually it is only explained with the binomial theorem.
July 9, 2023
This was really fun and made me love math
November 29, 2023
Want read this book heard lot about it
March 9, 2024
Great problem solving book that focuses on understanding rather than memorizing.
September 23, 2025
Practical and clear, with well picked examples that build on each other in a logical way. The detailed solutions make the problems easy to understand and approachable.
June 12, 2015
Starting with basics of counting arithmetic and build on top the concepts of permutation , combination , discrete & continuous probability in very easy to understand methods of explanation. He keeps pushing throughout the text not to memorize and try to understand the concept. He closes the book with combining algebra and counting arithmetic with Bernoulli's theorem and Pascal's identity and also throws in some famous puzzles and trivia about mathematicians every chapter. This would be a good book for your kids to teach the above concepts in a simple way
Displaying 1 - 11 of 11 reviews