Tuition increases on November 15th - Enroll soon to lock in current pricing!

Need Help?

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

Click here to Ask AoPS!

Intermediate Counting & Probability

Topics in discrete mathematics, including clever one-to-one correspondences, principle of inclusion-exclusion, generating functions, distributions, the pigeonhole principle, induction, constructive counting and expectation, combinatorics, systems with states, recursion, conditional probability, and introductory graph theory.

18 weeks

Diagnostics

ARE YOU READY? DO YOU NEED THIS?

Documents

SYLLABUS
18 weeks ARE YOU READY? DO YOU NEED THIS? SYLLABUS  

Schedule

Tuesday
Sep 17 - Feb 4
7:30 - 9:00
PM ET
Sep 17 - Feb 4
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
$560 (~$32/lesson)
$616 w/books
$560 (~$32/lesson)
With Books $616
CLOSED
Thursday
Nov 7 - Mar 27
7:30 - 9:00
PM ET
Nov 7 - Mar 27
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
$560 (~$32/lesson)
$616 w/books
$560 (~$32/lesson)
With Books $616
ENROLL
Monday
Feb 10 - Jun 16
7:30 - 9:00
PM ET
Feb 10 - Jun 16
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
$560 (~$32/lesson)
$616 w/books
$560 (~$32/lesson)
With Books $616
ENROLL
Sunday
Mar 23 - Aug 3
7:30 - 9:00
PM ET
Mar 23 - Aug 3
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
$560 (~$32/lesson)
$616 w/books
$560 (~$32/lesson)
With Books $616
ENROLL

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?

Students should have a complete mastery of basic counting as described in the diagnostic test above before taking this course. Students should also have a solid algebra background through our Intermediate Algebra class (or a typical honors Algebra 2 class and some Precalculus). Students who have completed the Art of Problem Solving Intermediate Algebra and Introduction to Counting & Probability classes should feel comfortable taking this class. (However, students are not required to take these classes before taking Intermediate Counting & Probability - use the diagnostic test above to determine if this class is right for you.)

Lessons

1 Review of Counting and Probability Basics
2 Principle of Inclusion & Exclusion
3 Advanced Inclusion & Exclusion
4 Constructive Counting
5 One-to-one Correspondences
6 The Pigeonhole Principle
7 Constructive Expectation
8 Distributions
9 Mathematical Induction and Fibonacci Numbers
10 Recursion and Catalan Numbers
11 Conditional Probability
12 Combinatorial Identities
13 Events with States
14 Generating Functions (Part 1)
15 Generating Functions (Part 2)
16 Graph Theory (Part 1)
17 Graph Theory (Part 2)
18 Bonus Topics and Challenging Problems

Required Textbook

Intermediate Counting & Probability
By David Patrick
An intermediate textbook in counting and probability for students in grades 9-12, containing topics such as inclusion-exclusion, recursion, conditional probability, generating functions, graph theory, and more.
Related course: Intermediate Counting and Probability

I enjoyed this class very much and learned quite a lot. Before I took the course, I was having a lot of difficulty with counting and probability, especially with distributions and expected value. Even the basic concepts were confusing to me at times, but over and over again throughout the course, the instructor's explanations and the homework assignments would bring the material into focus for me. I'm extremely glad I took the course, because the textbook alone wasn't enough.