gives the multinomial coefficient .


  • 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.


open allclose 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  (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 6 TemplateBox[{{z, +, 6}, z}, Binomial]:

Plot the imaginary part of 6 TemplateBox[{{z, +, 6}, z}, Binomial]:

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 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:

Wolfram Research (1988), Multinomial, Wolfram Language function,


Wolfram Research (1988), Multinomial, Wolfram Language function,


Wolfram Language. 1988. "Multinomial." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1988). Multinomial. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_multinomial, author="Wolfram Research", title="{Multinomial}", year="1988", howpublished="\url{}", note=[Accessed: 13-June-2024 ]}


@online{reference.wolfram_2024_multinomial, organization={Wolfram Research}, title={Multinomial}, year={1988}, url={}, note=[Accessed: 13-June-2024 ]}