Combinatorics

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Fundamental Counting Principles

"How do I love thee? Let me count the ways!" As choices increase in number, counting techniques are fundamental in determining the number of possible outcomes and orders. Explore the principles that form the basis of combinatorics.

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Multiplication and Addition PrinciplesFree

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Multiplication principlesFree

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Addition principlesFree

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Permutations and CombinationsFree

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PermutationsFree

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CombinationsFree

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Cards, Dice, and Lotteries

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Cards

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Dice

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Lotteries

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Permutations and Combinations With Repetitions

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Combinations with repetitions

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Binomial Coefficients

In situations where order is not important, binomial coefficients are used to calculate the number of possible combinations given a set number of choices. Begin practicing basic combinations and counting techniques.

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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Binomial Theorem

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Pascal’s Triangle

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Combinatorial Identities

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Multinomial Theorem

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The Pigeonhole Principle

A useful tool in combinatorics, the Pigeonhole Principle lets us assume that if $m$ items are placed in $n$ categories, and $m > n$, some categories must contain more than one item. Apply the consequences of this simple, but powerful, principle.

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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Basic Form of the Pigeonhole Principle

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Strong Form of the Pigeonhole Principle

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Distribution Problems

Utilize different techniques of distributing distinct or identical objects into boxes. Continue by exploring partitions and compositions of integers.

CompletionAccuracy

Accuracy is based on your most recent attempt.

Status

Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Distribution of Objects to Boxes

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Partitions and Compositions

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Partitions

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Compositions

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Stirling Numbers

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Inclusion-Exclusion

Take Venn Diagrams to the next level through examining the cardinality of the unions and intersections of sets.

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Accuracy is based on your most recent attempt.

Status

Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

The Cardinality of the Union of Sets

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The Cardinality of the Intersection of Set Complements

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Other Combinatorics Topics

Explore the various topics in combinatorics that don't fall 'discretely' into another theme: Latin Squares, Coding Theory, Ramsey Numbers, and more!

CompletionAccuracy

Accuracy is based on your most recent attempt.

Status

Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Latin Squares

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Block Designs

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Coding Theory

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