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Introduction to Counting & Probability Self-Paced

Summary

In this online class, students dive into the fundamentals of counting and probability, including casework, multiplication, permutations, combinations, Pascal's triangle, geometric probability, combinatorial identities, and the Binomial Theorem.

This advanced math course is offered in two formats: a live online course or a self-paced course. Both formats include instructor feedback, office hours, and a class message board for student support.

Introduction to Counting & Probability: Self-Paced

Length: 50 Lessons

In our self-paced Introduction to Counting & Probability course, students explore new concepts independently through a series of interactive, adaptive mathematical conversations.

Instead of prescheduled meeting times, self-paced math classes offer live interaction through the class message board, where students can connect with peers on challenging problems and receive support from instructors. We reinforce lessons with examples from the Introduction to Counting & Probability textbook, engaging videos, and various types of homework problems.

Although many students complete Introduction to Counting & Probability in 3–6 months, we offer unlimited class access for 9 months.

$400 $449 w/books

What's Included?

50 interactive lessons featuring lively, adaptive mathematical conversations to help students deepen their understanding
More than 80 supplementary videos from AoPS founder Richard Rusczyk, providing expert insights and guidance
Scheduling tools that allow students to set their preferred study hours each week, ensuring that activities are aligned with their personal schedule
Class message board where students can collaborate with peers, ask questions, and receive support from instructors and teaching assistants outside of class time
Periodic homework assignments including challenging short-answer problems, assigned readings from our Introduction to Counting & Probability textbook, and personalized practice through our adaptive online learning system, Alcumus
Detailed instructor evaluation of students' written solutions to open-response problems, designed to enhance mathematical writing and problem-solving skills
Comprehensive reporting that allows students and parents to track progress and performance throughout the course

Why AoPS?

Innovative Interactive Instruction

In our self-paced courses, students progress through interactive lessons that engage them in mathematical conversations, guiding them step-by-step and addressing common misconceptions. Students learn in various ways, so we offer multiple learning avenues: interactive lessons, a textbook, videos, diverse homework problems, and opportunities to engage in discussions with instructors and peers on the class message board.

Active Learning

Learning is not a spectator sport, and students learn best by tackling hard problems themselves. Our lessons start with challenging questions to help students develop new mathematical strategies. If students get stuck, our learning technology supplies helpful hints and adapts lesson structure to further reinforce key concepts.

Focus on Non-Routine, Challenging Problems

To prepare our students for the challenges of tomorrow, we teach them how to apply fundamentals to different types of problems, not just variations on the same problem they’ve seen before. Building a critical problem-solving skill set, our students are prepared for the rigors of top-tier colleges and internationally competitive careers.

Who Should Take This Class?

Before taking this course, students should complete a basic algebra course, including linear equations and multi-variable expressions. Students who have completed Introduction to Algebra A or equivalent will have the relevant background to get started.

We strongly recommend this course for students interested in participating in math competitions such as MATHCOUNTS and the AMC 8/10/12. Introduction to Counting & Probability dives deep into ideas that are essential for contest success — and exploring them in greater depth might just be their next best challenge. Outside of math competitions, discrete math is the math of modern computing and helpful for STEM careers.

Self-Paced vs. Weekly Live

Our self-paced class is designed for students who prefer to set their own study schedule. In this format, students can engage with peers and receive support from staff through the class message board. For students interested in a live group experience with real-time interactions in an online classroom setting, we recommend considering the weekly live version of this course.

Chapters

Show All Topics
Chapter 1
Counting is Arithmetic
  • Counting Lists of Numbers
  • Counting with Addition and Subtraction
  • Permutations
Chapter 2
Basic Counting Techniques
  • Casework
  • Complementary Counting
  • Constructive Counting
  • Counting with restrictions
Chapter 3
Correcting for Overcounting
  • Permutations with Repeated Elements
  • Counting Pairs of Items
  • Counting with Symmetries
Chapter 4
Committees and Combinations
  • Combinations
  • Combinatorial Identities
Chapter 5
More with Combinations
  • Paths on a Grid
  • Distinguishability
Chapter 6
Some Harder Counting Problems
  • Applications of Chapters 1–5
Chapter 7
Introduction to Probability
  • Definition of Probability
  • Counting Techniques in Probability
Chapter 8
Basic Probability Techniques
  • Probability and Addition
  • Complementary Probability
  • Probability and Multiplication
  • Probability and Dependent Events
Chapter 9
Think About It!
  • Using Symmetry in Problem-Solving
Chapter 10
Geometric Probability
  • Probability Using Lengths
  • Probability Using Areas
Chapter 11
Expected Value
  • Definition of Expected Value
  • Problem-Solving with Expected Value
Chapter 12
Pascal's Triangle
  • Constructing Pascal's Triangle
  • Pascal's Triangle as Combinations
  • More Combinatorial Identities
Chapter 13
The Hockey Stick Identity
  • Distributions
  • Sticks and Stones
  • The Hockey Stick Identity
Chapter 14
The Binomial Theorem
  • Proving the Binomial Theorem
  • Applying the Binomial Theorem to Problems
  • The Binomial Theorem in Identities
Chapter 15
More Challenging Problems
  • Applications of Chapters 7–14

Required Textbook

Introduction to Counting & Probability By David Patrick
A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more.
Related course: Introduction to Counting and Probability
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$400 $449 w/books

I cannot express how thankful I am for this course. I LOVED it! I had so much fun, and I always looked forward to Fridays to have class. The homework was challenging, but that is when it's the most fun! The instructor was a great teacher who kept me engaged in class, and we did cool problems in class. I learned so much, and I just thoroughly enjoyed this class. Thanks again!