Combinatorics - Wolfram|Alpha Examples

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Combinatorics

Combinatorics is a branch of mathematics dealing primarily with combinations, permutations and enumerations of elements of sets. It has practical applications ranging widely from studies of card games to studies of discrete structures. Wolfram|Alpha is well equipped for use analyzing counting problems of various kinds that are central to the field.

Factorials & Combinations

Work with factorials, binomial coefficients and related concepts.

Do computations with factorials:

100!12! / (4! * 6! * 2!)

Compute binomial coefficients (combinations):

30 choose 18

Compute a multinomial coefficient:

multinomial(3,4,5,8)

Evaluate a double factorial binomial coefficient:

DFBC(12, 6)DFBC(n, n-1)

Integer Partitions

Compute or count the partitions of an integer. Add constraints, specifying the number of parts or part size.

Compute the partitions of an integer:

integer partitions of 10

Specify a constraint on the number of parts:

partitions of 38 into 6 partsodd partitions of 14

Restrict to partitions into distinct parts:

partition 84 into distinct parts

Compute the number of partitions:

number of partitions of 1250

Combinatorial Functions

Learn about and do computations with combinatorial functions.

Compute a Bernoulli number:

Bernoulli B(16)

Compute a Stirling number:

S1(8, 4)

Compute a Frobenius number:

Frobenius number {4, 7, 12}

Compute Catalan numbers:

Catalan number

Compute Clebsch–Gordan coefficients:

Clebsch-Gordan calculator

Compute Wigner coefficients:

wigner 3jwigner 6jPermutations ›

Compute, count or do algebra with permutations of a set.

Compute the permutations of a set:

permutations of {a, b, c, d}

Count permutations:

number of permutations of 23 elements

Do algebra with permutations:

perm (1 2 3 4)^3(1 2 3)^-1More examples

Integer Compositions

Compute or count the compositions of an integer. Put constraints, specifying the number of parts or part size.

Compute the compositions of an integer:

compositions of 33

Specify a constraint on the parts:

compositions of 12 into even parts

Latin Squares

Get information about, compute or count Latin squares.

Get information about Latin squares:

Latin squares

Compute the number of Latin squares of a specified size:

6x6 Latin squares

Count normalized Latin squares:

normalized Latin squares of order 5

Compute bounds on the number of large-size Latin squares:

size 89 Latin square

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Step-by-Step Solutions for Discrete Mathematics

RELATED EXAMPLES

Graph Theory

Integers

Logic & Set Theory

Probability

Enumeration Problems

Solve a large variety of enumeration problems (also known as counting problems).

Count the number of possible colored bracelets:

bracelets 7 beads, 4 colors

Count the number of possible fixed necklaces:

fixed necklaces with 6 beads and 3 colors

Count Lyndon words:

5-ary Lyndon words of length 12