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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 (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:

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