Home Courses Premium Formula Sheets About

Loading...

Stay sharp. Stay curious.

Create a free account to save your progress, unlock every formula sheet, and keep your streak.

Create Free Account

Axiom AcademyBy BriTheMathGuy

Making math accessible and enjoyable through interactive lessons, engaging explanations, and a passion for teaching.

Learn

All Courses
Formula Sheets
Achievements

Test Prep

SAT Math
ACT Math
GRE Quant
AP Calculus

Company

About
Contact

Legal

Privacy Policy
Terms of Service
Accessibility

© 2026 Axiom Academy. All rights reserved.

Built with love for math.

Home
Courses
Combinatorics

n!

Combinatorics

Counting, permutations, combinations, and discrete structures

160Modules

9Units

Being rebuilt

This course is still in our original format. All the content is here and free to use, but it has not yet been through the rebuild that gives our newer courses their interactive animations and cleaner visuals — so you may hit some rough edges. Spot an error? Send us a note and we will fix it.

Recommended preparation

Comfort with Algebra 1 is enough to start; a taste of proof-writing (Intro to Proofs) helps with the trickier counting.

Algebra 1 Intro to Proofs

The Art of CountingINTRO

Why Combinatorics MattersINTRO

The Addition PrincipleLESSON

The Multiplication PrincipleLESSON

Basic Counting ExamplesEXAMPLE

Counting ExplorerACTIVITY

Password CombinationsREAL WORLD

The Subtraction PrincipleLESSON

The Division PrincipleLESSON

Counting Principles PracticePRACTICE

Counting StrategiesINTRO

The Bijection PrincipleLESSON

Double CountingLESSON

Counting Strategy ExamplesEXAMPLE

Strategy PracticePRACTICE

Counting Principles SummarySUMMARY

Counting Principles ReferenceREFERENCE

Counting Principles QuizQUIZ

Arranging ObjectsINTRO

When Order MattersINTRO

Permutation DefinitionLESSON

Basic Permutation ExamplesEXAMPLE

Permutation GeneratorACTIVITY

Factorials and n!LESSON

r-Permutations P(n,r)LESSON

Permutations PracticePRACTICE

Circular PermutationsLESSON

Circular Permutations PracticePRACTICE

Permutations with RepetitionLESSON

Multiset PermutationsLESSON

DerangementsLESSON

Advanced Permutation ExamplesEXAMPLE

Derangement CounterACTIVITY

Permutation Groups PreviewLESSON

Seating ArrangementsREAL WORLD

Rankings and OrdersREAL WORLD

Permutations SummarySUMMARY

Permutations ReferenceREFERENCE

Permutations QuizQUIZ

The Art of ChoosingINTRO

When Order Doesn't MatterINTRO

Combination DefinitionLESSON

Basic Combination ExamplesEXAMPLE

Binomial CoefficientsLESSON

Combination CalculatorACTIVITY

Combination PropertiesLESSON

Combinations with RepetitionLESSON

Combinations PracticePRACTICE

Stars and Bars MethodLESSON

Stars and Bars ExamplesEXAMPLE

Stars and Bars PracticePRACTICE

Committee ProblemsLESSON

Distributing ObjectsLESSON

Lattice PathsLESSON

Lattice Path VisualizerACTIVITY

Lottery OddsREAL WORLD

Poker HandsREAL WORLD

Combinations SummarySUMMARY

Combinations ReferenceREFERENCE

Combinations QuizQUIZ

Expanding PowersINTRO

Pascal's TriangleINTRO

The Binomial TheoremLESSON

Binomial DistributionsREAL WORLD

Binomial Expansion ExamplesEXAMPLE

Binomial ExpanderACTIVITY

Binomial Theorem PracticePRACTICE

Proving the Binomial TheoremLESSON

Pascal's Triangle PropertiesLESSON

Pascal's Triangle ExplorerACTIVITY

Pascal's IdentityLESSON

Identity Proof ExamplesEXAMPLE

Identity PracticePRACTICE

Binomial IdentitiesLESSON

Vandermonde's IdentityLESSON

Multinomial TheoremLESSON

Negative Binomial CoefficientsLESSON

Binomial Theorem SummarySUMMARY

Binomial Theorem ReferenceREFERENCE

Binomial Theorem QuizQUIZ

The Pigeonhole IdeaINTRO

Proving Something ExistsINTRO

Basic Pigeonhole PrincipleLESSON

Pigeonhole Problem ExamplesEXAMPLE

Pigeonhole VisualizerACTIVITY

Pigeonhole PracticePRACTICE

Generalized Pigeonhole PrincipleLESSON

Classic ApplicationsLESSON

Introduction to Ramsey TheoryLESSON

The Party ProblemLESSON

Ramsey Theory ExamplesEXAMPLE

Ramsey Theory PracticePRACTICE

The Birthday ProblemREAL WORLD

Erdős-Szekeres TheoremLESSON

Pigeonhole SummarySUMMARY

Pigeonhole ReferenceREFERENCE

Pigeonhole Principle QuizQUIZ

Counting Overlapping SetsINTRO

Venn Diagram CountingINTRO

Two-Set FormulaLESSON

Three-Set FormulaLESSON

General Inclusion-ExclusionLESSON

Inclusion-Exclusion ExamplesEXAMPLE

Inclusion-Exclusion PracticePRACTICE

Derangements via I-ELESSON

Euler's Phi via I-ELESSON

Counting SurjectionsLESSON

Surjection ExamplesEXAMPLE

Surjection PracticePRACTICE

Sieve MethodsLESSON

Survey AnalysisREAL WORLD

I-E CalculatorACTIVITY

Venn Diagram BuilderACTIVITY

Inclusion-Exclusion SummarySUMMARY

Inclusion-Exclusion ReferenceREFERENCE

Inclusion-Exclusion QuizQUIZ

Counting RecursivelyINTRO

The Fibonacci SequenceINTRO

Recurrence Relations DefinedLESSON

Recurrence SolverACTIVITY

Linear Homogeneous RecurrencesLESSON

Characteristic Equation MethodLESSON

Solving Recurrence ExamplesEXAMPLE

Repeated RootsLESSON

Nonhomogeneous RecurrencesLESSON

Recurrence PracticePRACTICE

Binet's FormulaLESSON

Population Growth ModelsREAL WORLD

Catalan NumbersLESSON

Catalan Number ExamplesEXAMPLE

Catalan Number VisualizerACTIVITY

Stirling NumbersLESSON

Special Sequences PracticePRACTICE

Algorithm AnalysisREAL WORLD

Recurrence SummarySUMMARY

Recurrence ReferenceREFERENCE

Recurrence Relations QuizQUIZ

Power Series for CountingINTRO

Polynomial MagicINTRO

Ordinary Generating FunctionsLESSON

OGF Problem ExamplesEXAMPLE

Generating Function CalculatorACTIVITY

OGF PracticePRACTICE

OGF OperationsLESSON

Solving Recurrences with OGFsLESSON

Exponential Generating FunctionsLESSON

EGF Problem ExamplesEXAMPLE

EGF PracticePRACTICE

EGF ApplicationsLESSON

Partition FunctionsLESSON

Extracting CoefficientsLESSON

Snake Oil MethodLESSON

Coin Change ProblemREAL WORLD

Generating Functions SummarySUMMARY

Generating Functions ReferenceREFERENCE

Generating Functions QuizQUIZ

Problem Solving GuideREFERENCE

Combinatorics Master ReferenceREFERENCE

Complete Formula SheetFORMULA SHEET

Comprehensive Combinatorics TestQUIZ