Multinomial[n1,n2,…]
gives the multinomial coefficient .


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 allBasic Examples (5)
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)
The precision of the output tracks the precision of the 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:
Find a value of n for which Multinomial[n,1,1]=15:
Visualization (2)
Plot the Multinomial as a function of its parameter:
Function Properties (11)
The real domain of as a function of its last parameter
:
Approximate function range of :
is neither non-decreasing nor non-increasing:
is neither non-negative nor non-positive:
does not have either singularity or discontinuity:
is neither convex nor concave:
TraditionalForm formatting:
Differentiation (2)
Series Expansions (2)
Find the Taylor expansion using Series:
Generalizations & Extensions (1)
Multinomial threads elementwise over lists:
Applications (4)
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)
Nested multinomials over the complex plane:
Plot Multinomial for complex arguments:
Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0)
Text
Wolfram Research (1988), Multinomial, Wolfram Language function, https://reference.wolfram.com/language/ref/Multinomial.html.
CMS
Wolfram Language. 1988. "Multinomial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Multinomial.html.
APA
Wolfram Language. (1988). Multinomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Multinomial.html
BibTeX
@misc{reference.wolfram_2025_multinomial, author="Wolfram Research", title="{Multinomial}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Multinomial.html}", note=[Accessed: 11-August-2025]}
BibLaTeX
@online{reference.wolfram_2025_multinomial, organization={Wolfram Research}, title={Multinomial}, year={1988}, url={https://reference.wolfram.com/language/ref/Multinomial.html}, note=[Accessed: 11-August-2025]}