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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.
What's Included?
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.
- To determine if you’re ready for the course, students can take the diagnostic pretest.
- To determine if you need this course, students can take the diagnostic post-test.
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.
- Counting Lists of Numbers
- Counting with Addition and Subtraction
- Permutations
- Casework
- Complementary Counting
- Constructive Counting
- Counting with restrictions
- Permutations with Repeated Elements
- Counting Pairs of Items
- Counting with Symmetries
- Combinations
- Combinatorial Identities
- Paths on a Grid
- Distinguishability
- Applications of Chapters 1–5
- Definition of Probability
- Counting Techniques in Probability
- Probability and Addition
- Complementary Probability
- Probability and Multiplication
- Probability and Dependent Events
- Using Symmetry in Problem-Solving
- Probability Using Lengths
- Probability Using Areas
- Definition of Expected Value
- Problem-Solving with Expected Value
- Constructing Pascal's Triangle
- Pascal's Triangle as Combinations
- More Combinatorial Identities
- Distributions
- Sticks and Stones
- The Hockey Stick Identity
- Proving the Binomial Theorem
- Applying the Binomial Theorem to Problems
- The Binomial Theorem in Identities
- Applications of Chapters 7–14
Required Textbook
Related course: Introduction to Counting and Probability