Combinatorics: Counting Rules & Formulas

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Combinatorics Visual Tools

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Interactive explorers for every standard counting scenario: full, partial, circular and repetition permutations, combinations, partitions, distributions, weak and strong compositions, and the Pascal triangle — each one animated and adjustable so the formula and the arrangement it counts stay visibly connected.

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Circular Permutation Visualizer

Watch (n-1)! circular permutations build one by one — one item anchored at the top breaks the rotational symmetry while the remaining items cycle through every distinct seating around the circle. Adjust n and step through the build animation.

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Key Terms

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Key Terms

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Full Permutation Visualizer

Build every n! permutation of n distinct items one ball at a time. Each arrangement starts with a different first item, and rows group all permutations sharing that starter — so you literally see why n! splits as n × (n−1)!

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Key Terms

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Key Terms

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Pascal

Click any cell to focus a binomial coefficient C(n, r) and switch between four identity modes — Pascal

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Permutation with Identical Items Visualizer

Cycle through six multisets — AAB, AAAB, AABB, AABC, AAABB, AABBC — and watch every distinct permutation build one position at a time. Only the specific copy in use dims in the source row, exposing why n! divides by k! for each repeated group.

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Combinatorics Formulas Reference

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Combinatorics formulas reference: factorial, permutation P(n,r), combination C(n,r), binomial coefficient, multinomial, stars and bars, Pascal

Counting Principles6

Addition RuleRead more →Multiplication RuleRead more →Complementary CountingRead more →Double CountingRead more →Pigeonhole PrincipleRead more →Pigeonhole Principle (Generalized)Read more →

Inclusion Exclusion5

Permutations9

Combinations & Distributions5

Binomial Coefficient Identities10

Binomial & Multinomial Theorems7

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Combinatorics Terms and Definitions

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Combinatorics glossary with 22 defined terms: counting principles, permutations, combinations, distributions, binomial coefficient, and Pascal

Counting Principles6

Addition RuleThe addition rule states that if a count splits into

k

mutually exclusive cases with

m_1, m_2, \ldots, m_k

outcomes each, the total number of…Read more →Multiplication RuleThe multiplication rule states that if an outcome is built from

k

independent steps with

m_1, m_2, \ldots, m_k

options each, the combined…Read more →Complementary CountingComplementary counting computes the size of a set by subtracting the size of its complement from the universe: if

U

is the universe and…Read more →Double CountingDouble counting is a proof technique in which the same set is enumerated by two different strategies; the resulting expressions both equal the size…Read more →Pigeonhole PrincipleThe pigeonhole principle states that if

n

items are distributed among

k

containers and

n > k

, then at least one container holds at least two…Read more →Inclusion-Exclusion PrincipleThe inclusion-exclusion principle computes the size of a union of

n

sets by alternately adding the sizes of

k

-fold intersections:…Read more →

Permutations8

FactorialThe factorial of a non-negative integer

n

, written

n!

, is the product of all positive integers from

1

up to

n

n

distinct items in a linear sequence, with each item appearing exactly once. The number of full…Read more →Partial PermutationA partial permutation is a selection of

r

distinct items from

n

available items followed by their arrangement into a linear sequence, with no…Read more →Permutation with RepetitionA permutation with repetition is an arrangement of

r

positions where each position is filled independently from

n

available items, with the same…Read more →Permutation with Identical ItemsA permutation with identical items is an arrangement of

n

objects in a linear sequence where some objects are indistinguishable from one another,…Read more →Circular PermutationA circular permutation is an arrangement of

n

distinct items around a circle, where two arrangements are considered identical if one is a rotation…Read more →DerangementA derangement is a permutation of a set in which no element appears in its original position. The number of derangements of an

n

-element set,…Read more →

Combinations & Distributions5

CombinationA combination is a selection of items from a collection where order does not matter. The number of ways to select

r

items from

n

distinct items…Read more →Partition into GroupsA partition into groups divides

n

distinct items into

k

unlabeled subsets where only the grouping matters, not the order within groups or names…Read more →Weak CompositionA weak composition is a distribution of

n

identical items into

r

labeled containers where some containers may remain empty. The number of weak…Read more →Strong CompositionA strong composition is a distribution of

n

identical items into

r

labeled containers where every container must receive at least one item. The…Read more →Distribution into CellsA distribution into cells assigns each of

n

distinct items to one of

r

labeled containers, producing a mapping from items to containers. The…Read more →

Binomial Structure3

Binomial CoefficientThe binomial coefficient

\binom{n}{k}

, read "

n

choose

k

", is the number of

k

-element subsets of an

n

-element set. For non-negative integers…Read more →Multinomial CoefficientThe multinomial coefficient counts the number of ways to partition

n

distinct items into

r

labeled groups of specified sizes…Read more →Pascal's TrianglePascal's triangle is the triangular array of binomial coefficients in which row

n

(indexed from

0

) contains the

n + 1

values…Read more →

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Combinatorics Basics

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The complete counting toolkit on one page: the five counting principles, the inclusion-exclusion correction for overlapping sets, permutations versus combinations, the ten standard counting scenarios with their formulas, and the binomial coefficient with the binomial theorem — plus the bridges into probability, set theory, and algebra.

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Binomial Coefficient

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Binomial Theorem

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Binomial theorem for expanding (a+b)^n: general term formula, special cases, multinomial generalization, combinatorial proof, and worked example expansions.

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Combinatorics Calculator

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Calculate the number of ways to arrange

n

distinct objects in a sequence.

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Combinations

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Combinations in combinatorics: simple combinations, partition into groups, weak and strong composition, and distribution into cells, with formulas and examples.

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Counting Principles in Combinatorics

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Counting principles of combinatorics: addition rule, multiplication rule, complementary counting, double counting, and the pigeonhole principle explained.

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Inclusion-Exclusion Principle

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Inclusion-exclusion principle for counting unions of overlapping sets. Two-set, three-set, and general n-set formulas with derangement and surjection examples.

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Permutations

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Permutations in combinatorics: full, partial, with repetition, with identical items, circular, and derangements. Formulas, notation, and worked examples.

Permutations Visualizer

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