We have your learning goals covered with Spring and Summer courses available. Enroll today!

Need Help?

Need help finding the right class? Have a question about how classes work?

Click here to Ask AoPS!

Introduction to Counting & Probability

Summary

In this online class, students dive into the fundamentals of counting and probability, including casework, multiplication, permutations, combinations, Pascal's triangle, 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: Weekly Live

Length: 12 Weeks

Our live Introduction to Counting & Probability course brings together advanced problem solvers from around the world for 90-minute classes. Experienced instructors lead students to creatively problem-solve with casework, permutations, combinations, Pascal's triangle, combinatorial identities, and the Binomial Theorem.

Students refer to our Introduction to Counting & Probability textbook and videos, while honing their skills on various types of engaging homework problems. They can also use the class message board to collaborate with their peers and reach out to instructors outside of class.

Live Online Course Features

12 live class sessions taught by our outstanding faculty in an interactive text- and image-based classroom designed for collaboration and motivated students
More than 80 supplementary videos by AoPS founder Richard Rusczyk
Detailed instructor evaluation of students' written solutions to open-response problems designed to strengthen mathematical writing skills
Class message board where students can collaborate with their classmates or ask instructors and AoPS staff questions outside of class time
Weekly homework assignments with challenging short-answer problems, assigned readings, and practice on our adaptive online learning system, Alcumus
Comprehensive reporting that allows students and parents to thoroughly track progress and performance throughout the course

Why AoPS?

Extraordinary Live Instruction from Accomplished Faculty

Our teachers were outstanding math students themselves, and many were winners of major math awards as well as recipients of degrees from schools like MIT, Harvard, Stanford, and Princeton. Our instructors understand from their personal experiences how to inspire and challenge eager learners, and they serve as role models for our students who aspire to similar success.

Active Learning

Students learn best by actively engaging with challenging problems. Our lessons start with thought-provoking questions, encouraging independent insights and problem-solving strategies. Instructors offer hints and guidance as needed and reinforce key concepts after problems are solved. To support diverse learning styles, we provide live classes, textbooks, videos, homework problems, and opportunities for discussion with instructors and peers.

Focus on Non-Routine, Challenging Problems

In today's world, many jobs previously requiring routine problem-solving are now handled by computers. To prepare students for the future, we focus on teaching them how to apply core principles to new, non-routine problems. This approach ensures that students develop essential skills for top-tier colleges and competitive careers, preparing them for lifelong learning rather than rote memorization.

Schedule: Introduction to Counting & Probability

The current and upcoming sections of this course are listed below. All sections cover the same course material.

Sunday
Feb 9 - Apr 27
3:30 - 5:00
PM ET  
Feb 9 - Apr 27
3:30 - 5:00 PM Eastern
2:30 - 4:00 PM Central
1:30 - 3:00 PM Mountain
12:30 - 2:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
CLOSED
Sunday
Mar 16 - Jun 8
7:30 - 9:00
PM ET
Mar 16 - Jun 8
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Wednesday
Apr 16 - Jul 2
7:30 - 9:00
PM ET
Apr 16 - Jul 2
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Thursday
May 15 - Jul 31
7:30 - 9:00
PM ET
May 15 - Jul 31
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Sunday
Jun 1 - Aug 24
7:30 - 9:00
PM ET
Jun 1 - Aug 24
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Thursday
Jun 12 - Aug 28
7:30 - 9:00
PM ET
Jun 12 - Aug 28
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Wednesday
Jul 2 - Sep 17
7:30 - 9:00
PM ET
Jul 2 - Sep 17
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Sunday
Jul 27 - Oct 19
7:30 - 9:00
PM ET
Jul 27 - Oct 19
7:30 - 9:00 PM Eastern
6:30 - 8:00 PM Central
5:30 - 7:00 PM Mountain
4:30 - 6:00 PM Pacific
Click here to see more time zones
$400 (~$34/lesson)
$449 w/books
$400 (~$34/lesson)
$449 w/books
ENROLL
Fall 2025This course will be offered in Fall 2025. Click here to join our mailing list to be notified when the course schedule is available.
Don't see a schedule option that works for you? Check out the self-paced version of this class.

AoPS Holidays

There are no classes May 24 ‐ 26, July 4 ‐ 6, August 29 ‐ September 1, October 31, November 24 ‐ November 30, and December 20, 2025 ‐ January 2, 2026.

Who Should Take This Class?

This course is appropriate for students in grades 6-9 who have mastered basic algebra up through solving linear equations and manipulating multi-variable expressions. Students who have completed our Introduction to Algebra A course should have sufficient background.

This course is roughly the same difficulty as our Introduction to Number Theory class. For those preparing for contests, this course should help with MATHCOUNTS and the AMC 8/10/12 tests.

Weekly Live vs. Self-Paced

Our weekly live class is designed for students who prefer a live group experience. In this format, students learn in an online classroom, engaging in real-time with their instructor and peers. For students interested in setting their own study schedule, we recommend considering the self-paced version of this course.

Lessons

Show All Topics
Lesson 1
Lists, Venn Diagrams, Addition, Multiplication
  • Counting Lists of Numbers
  • Counting with Addition and Subtraction
  • Permutations
Lesson 2
Casework, Constructions, and Restriction
  • Casework
  • Complementary Counting
  • Constructive Counting
  • Counting with Restrictions
Lesson 3
Overcounting and Combinations
  • Permutations with Repeated Elements
  • Counting Pairs of Items
  • Counting with Symmetries
  • Combinations
  • Combinatorial Identities
Lesson 4
Combinations and Distinguishability
  • Paths on a Grid
  • Distinguishability
Lesson 5
Challenging Problems Day
  • Applications of Lessons 1-4
Lesson 6
Introduction to Probability
  • Definition of Probability
  • Counting Techniques in Probability
Lesson 7
Probability and Arithmetic
  • Probability and Addition
  • Complementary Probability
  • Probability and Multiplication
  • Probability and Dependent Events
Lesson 8
Geometric Probability, Think About It!, and Expected Value
  • Using Symmetry in Problem-Solving
  • Probability Using Lengths
  • Probability Using Areas
  • Definition of Expected Value
  • Problem-Solving with Expected Value
Lesson 9
Pascal's Triangle and Identities
  • Constructing Pascal's Triangle
  • Pascal's Triangle as Combinations
  • More Combinatorial Identities
Lesson 10
Distributions and The Hockey Stick Identity
  • Distributions
  • Sticks and Stones
  • The Hockey Stick Identity
Lesson 11
The Binomial Theorem
  • Proving the Binomial Theorem
  • Applying the Binomial Theorem to Problems
  • The Binomial Theorem in Identities
Lesson 12
Challenging Problems Day 2
  • Applications of Lessons 6-11

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
View Details

I had so much fun in this class! It was BY FAR my favorite class this semester! I think it was very informative and really interesting and I would highly (very, very, very highly) recommend this class to anybody with no hesitation!