BUILT-IN MATHEMATICA SYMBOL

Multinomial

gives the multinomial coefficient

.

MORE INFORMATION

Integer mathematical function, suitable for both symbolic and numerical manipulation.

The multinomial coefficient Multinomial, 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 (1)

The 1, 2, 1 multinomial coefficient appears as the coefficient of

:

In[1]:=

Out[1]=

The 1, 2, 1 multinomial coefficient appears as the coefficient of

:

In[2]:=

Out[2]=

Scope (8)

Special cases evaluate for symbolic arguments:

Evaluate with any number of arguments:

Evaluate for large integer arguments:

Evaluate for half-integer arguments:

Numerical generalization:

Evaluate for complex arguments:

Series expansion at a generic point:

TraditionalForm formatting:

Generalizations & Extensions (1)

Multinomial threads element-wise 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: