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Introduction to Counting & Probability

Name: Introduction to Counting & Probability
Price: 400 USD

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.

Diagnostics: Are You Ready? Do You Need This Course?

Required Text: Introduction to Counting & Probability

Weekly Live Online Class Self-Paced Online Class

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.

Why AoPS? Schedule Who Should Take? Topics Required Textbook

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

Dana Paquin

Stanford PhD Budapest Semesters Invitee Stanford Math Circle Director

Brian Edwards

Northwestern PhD MOP Invitee USA Math Talent Search Director

Josh Zucker

Stanford & Berkeley MS Putnam Top 10 US Physics Team

Ashley Ahlin

UChicago PhD Westinghouse Winner MATHCOUNTS Top 3

Miles Dillon Edwards

MOP Invitee RSI Invitee Putnam Top 25

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.

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	Aaron Doman	$400 (~$34/lesson) $449 w/books	$400 (~$34/lesson) $449 w/books CLOSED
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	Luís Finotti	$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	Angel Bell	$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	Dana Paquin	$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	Asa Frank	$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	Charles Buehrle	$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	Sarah Seales	$400 (~$34/lesson) $449 w/books	$400 (~$34/lesson) $449 w/books ENROLL
Wednesday Jul 9 - Sep 24	7:30 - 9:00 PM ET Jul 9 - Sep 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	Heather Finotti	$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	Johann Edir Hernandez Ybarra	$400 (~$34/lesson) $449 w/books	$400 (~$34/lesson) $449 w/books ENROLL
Fall 2025	This 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.

Lessons

Show All Topics

Full Course Syllabus

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

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