# 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

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## 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(25)

### Numerical Evaluation(4)

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:

### 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:

### 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:

## 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:

## Possible Issues(3)

Large arguments can give results too large to be computed explicitly:  Machine-number inputs can give highprecision 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: