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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 of lively adaptative mathematical conversation
More than 80 supplementary videos by AoPS founder Richard Rusczyk
Class message board where students can ask instructors questions and collaborate outside of class time
Office hours held every day where instructors and assistants provide live homework help
Periodic homework assignments including challenging short-answer problems, assigned readings from our Introduction to Counting & Probability textbook, and practice on our adaptive online learning system, Alcumus
Detailed report that allows students and parents to track progress and performance throughout the course
Scheduling tools that let students specify how long they wish to work each day of the week so that we can assign activities appropriately
Detailed instructor evaluation of students' written solutions to open-response problems designed to strengthen mathematical writing skills

Why AoPS?

Innovative Interactive Instruction

In our self-paced courses, students work their way through carefully scripted interactive lessons that engage students in mathematical conversations. Each lesson reacts to student input, guiding them step by step through problems and tackling common misconceptions along the way. When students have questions, they can turn to the class message board where instructors provide additional assistance.

Focus on New, 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.

Multiple Learning Avenues

Students learn in many ways, so we teach through multiple avenues. In addition to the interactive lessons, our students can read the Introduction to Counting & Probability textbook, watch videos, solve different types of homework problems, and participate in math conversations with instructors and peers on the class message board.

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 wish to set their own schedule for their studies. While this course does provide some peer interaction and live teacher support through the class message board and office hours, students wishing to have a live group experience in an online classroom with instructors and peers should consider our 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

Overall, the course was excellent and I learned a lot. The textbook was awesome. The problems given in Alcumus and in the challenge set were fun to work on and helped me to understand the math concepts covered in the class. Each problem given was so unique (and sometimes even hilarious) that it tested my understanding of the concepts in completely different ways each time.