Multinomial

The multinomial coefficient $\frac{(k_1 + k_2 + \cdots)!}{k_1!\,k_2!\cdots}$ generalises Binomial to more than two parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaMultinomial theorem↗Multinomial coefficient↗MathWorldMultinomialCoefficient↗WikidataQ20820114↗Britannicatopic/multinomial-coefficient↗Wolfram LanguageMultinomial!↗SageMathmultinomial($*,)Juliaenumeratio_multinomial($*,)

Multinomial(k_1, k_2, ...)the multinomial coefficient for parts $k_1, k_2, \ldots$

Multinomial(k_1, k_2, …)The multinomial coefficient $\frac{(k_1 + k_2 + \cdots)!}{k_1!\,k_2!\cdots}$ generalises Binomial to more than two parts.

Typed signatures

Type	Package	Applies
(integer+) -> integer	analyticreplaces compute-engine	on Around, CenteredInterval, Interval
(integer+) -> integer	compute-engine	always
(any*) -> any	number-theoryreplaces analytic	always

Domain: Combinatorics

Details

• Counts the ways to split $k_1 + k_2 + \cdots$ labeled items into groups of the given sizes.
• With two arguments it reduces to Binomial: $\binom{k_1+k_2}{k_1}$.
• Orderless: permuting the arguments doesn't change the value.
• All-zero arguments and a single argument both reduce to the empty product, 1.
• compute-engine requires all-integer arguments; a list argument is threaded over element-wise, as Wolfram's Listable heads do.

Examples

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{Multinomial}(1, 2, 1)$Out$12$

live ↗

In$\mathrm{Multinomial}(2, 3, 4)$Out$1\,260$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

live ↗

In$\mathrm{Multinomial}(\bigl\lbrack2, 3, 5\bigr\rbrack, 3)$Out$\bigl\lbrack10, 20, 56\bigr\rbrack$

Zero parts contribute $0! = 1$

live ↗

In$\mathrm{Multinomial}(0, 0, 1)$Out$1$

Real arguments evaluate via the Gamma function, $\frac{\Gamma(8.2)}{2!\,\Gamma(1.2)\,5!}$

live ↗

In$\mathrm{Multinomial}(2, 0.2, 5)$Out$34.317\,803\,519\,999\,95$

Complex arguments evaluate via the Gamma function too

live ↗

In$\mathrm{Multinomial}(1+i, 0.2, 4)$Out$2.894\,060\,302\,449\,624+9.104\,625\,875\,691\,285i$

An Around argument propagates its uncertainty through the head as one function: $\frac{(a+2)(a+1)}{2}$ at $a = 2 \pm 0.01$ is $6 \pm 0.035$

live ↗

In$\mathrm{Multinomial}(\mathrm{Around}(2, 0.01), 2)$Out$\mathrm{Around}(6, 0.035)$

Applications #

The number of distinct letter arrangements of MISSISSIPPI: 1 M, 4 I's, 4 S's, 2 P's

live ↗

In$\mathrm{Multinomial}(1, 4, 4, 2)$Out$34\,650$

The coefficient of $a^2 b^3 c$ in $(a+b+c)^6$

live ↗

In$\mathrm{Multinomial}(2, 3, 1)$Out$60$

Properties #

With two arguments it reduces to Binomial: $\binom{n_1+n_2}{n_1}$

live ↗

In$\mathrm{Multinomial}(7, 3)=\binom{10}{3}$Out$\mathrm{True}$

Orderless: permuting the arguments doesn't change the value

live ↗

In$\mathrm{Multinomial}(1, 2, 3)=\mathrm{Multinomial}(3, 1, 2)$Out$\mathrm{True}$

The factorial form $\frac{(k_1+k_2+k_3)!}{k_1!\,k_2!\,k_3!}$. See Factorial

live ↗

In$\mathrm{Multinomial}(3, 4, 5)=\frac{12!}{3!×4!×5!}$Out$\mathrm{True}$

All-ones parts give $n!$: every part is a single labelled item

live ↗

In$\mathrm{Multinomial}(1, 1, 1, 1)$Out$24$

Possible issues #

All-zero arguments still give the empty product, 1

live ↗

In$\mathrm{Multinomial}(0, 0, 0)$Out$1$

A single argument is always 1: there's only one way to split n items into one group

live ↗

In$\mathrm{Multinomial}(5)$Out$1$

With no arguments it is the empty product, 1

live ↗

In$\mathrm{Multinomial}()$Out$1$

About

Examples	17
Checked against	Wolfram Language 16 agree1 differSageMath 13 agree3 errormathlib4 12 agree2 errorJulia 14 agree1 error
Crosswalk	Wikipedia ×2MathWorldWikidataBritannicaWolfram LanguageSageMathJulia

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Nat.multinomial over the indices of the argument list, so the empty call is 1; natural arguments only.

See also: Binomial, Factorial