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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: 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 several types of 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 one of 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
Office hours held every day, where instructors and assistants provide live homework help, and a class message board, where students can ask questions and collaborate
Weekly homework including short-answer questions and open-response writing problems, on which students receive detailed feedback
Automated reminders for upcoming classes, reading assignments, and homework and thorough progress tracking

Why AoPS?

Extraordinary Live Instruction from Accomplished Faculty

Our teachers were outstanding math students themselves. Many won major math awards and earned degrees from schools like MIT, Harvard, Stanford, and Princeton. They understand from personal experience what our students need, and they serve as role models for students who aspire to similar success.

The weekly class sessions are lively interactive discussions. Our online classroom allows all students to answer any instructor question and lets students ask questions at any time.

Schedule: Introduction to Counting & Probability

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

Wednesday
Sep 25 - Dec 18
7:30 - 9:00
PM ET
Sep 25 - Dec 18
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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
CLOSED
Sunday
Oct 27 - Feb 2
7:30 - 9:00
PM ET
Oct 27 - Feb 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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
CLOSED
Thursday
Dec 5 - Mar 6
7:30 - 9:00
PM ET
Dec 5 - Mar 6
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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
ENROLL
Wednesday
Jan 8 - Mar 26
7:30 - 9:00
PM ET
Jan 8 - Mar 26
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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
ENROLL
Thursday
Jan 30 - Apr 17
7:30 - 9:00
PM ET
Jan 30 - Apr 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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
ENROLL
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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
ENROLL
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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 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
$390 (~$33/lesson)
$439 w/books
$390 (~$33/lesson)
$439 w/books
ENROLL
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 November 25 ‐ December 1, December 21 ‐ January 3, May 24 ‐ 26, July 4 ‐ 6, August 29 ‐ September 1, and October 31, 2025.

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.

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

The course really taught me how to think in a problem-solving fashion, and applying previous knowledge to new concepts and proofs. I learned a lot about how to prove things and how to explain them in a logical manner. I think it was a great course to begin studies in discrete mathematics.