Wolfram Language & System Documentation Center

Multinomial

Multinomial[n₁,n₂,…]

gives the multinomial coefficient .

Details

Integer mathematical function, suitable for both symbolic and numerical manipulation.
The multinomial coefficient Multinomial[n₁,n₂,…], denoted , gives the number of ways of partitioning distinct objects into sets, each of size (with ).
Multinomial automatically threads over lists.

Examples

open all close all

Basic Examples (5)

Evaluate numerically:

The 1, 2, 1 multinomial coefficient appears as the coefficient of x y^2 z:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (27)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Multinomial function using MatrixFunction:

Specific Values (4)

Values of Multinomial at fixed points:

Multinomial for symbolic n:

Values at zero:

Find a value of n for which Multinomial[n,1,1]=15:

Visualization (2)

Plot the Multinomial as a function of its parameter:

Plot the real part of :

Plot the imaginary part of :

Function Properties (11)

The real domain of as a function of its last parameter :

The complex domain:

Approximate function range of :

has the mirror property:

is an analytic function of x:

is neither non-decreasing nor non-increasing:

is not injective:

is surjective:

is neither non-negative nor non-positive:

does not have either singularity or discontinuity:

is neither convex nor concave:

TraditionalForm formatting:

Differentiation (2)

The first derivative with respect to n₃:

Higher derivatives with respect to n₃:

Plot the higher derivatives with respect to n₃ when n₁=n₂=3:

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

Generalizations & Extensions (1)

Multinomial threads elementwise over lists:

Applications (4)

Illustrate the multinomial theorem:

Plot isosurfaces of the number of ways to put elements in three boxes:

Multinomial probability distribution:

Volume of a hyper-super-ellipsoid is :

Compare with direct integration:

Properties & Relations (4)

With two arguments, Multinomial gives binomial coefficients:

Use FullSimplify to simplify expressions involving multinomial coefficients:

Use FunctionExpand to expand into Gamma functions:

Multinomial is Orderless:

Possible Issues (3)

Large arguments can give results too large to be computed explicitly:

Machine-number inputs can give high‐precision results:

As a multivariate function, Multinomial is not continuous in all variables at negative integers:

Neat Examples (3)

Trinomials mod 2:

Modulo 3:

Nested multinomials over the complex plane:

Plot Multinomial for complex arguments:

Top