The multinomial coefficients
(1)
are the terms in the multinomial series expansion. In other words, the number of distinct permutations in a multiset
of
The multinomial coefficient is returned by the Wolfram Language function Multinomial n1 ,
n2 , ...].
The special case
(2)
where binomial coefficient .
The multinomial coefficients satisfy
and so on (Gosper 1972).
See also Ball Picking ,
Binomial Coefficient ,
Choose ,
Combination ,
Dyson's Conjecture ,
Multichoose ,
Multinomial Series ,
Multiset ,
Permutation ,
q -Multinomial
Coefficient,
String ,
Trinomial
Coefficient ,
Zeilberger-Bressoud Theorem
Related Wolfram sites http://functions.wolfram.com/GammaBetaErf/Multinomial/
Explore with Wolfram|Alpha
References Abramowitz, M. and Stegun, I. A. (Eds.). "Multinomial Coefficients." §24.1.2 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Gosper, R. W. Item 42 in
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA:
MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 16, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item42 . Skiena,
S. Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Spiegel, M. R. Theory
and Problems of Probability and Statistics. Referenced on Wolfram|Alpha Multinomial Coefficient
Cite this as:
Weisstein, Eric W. "Multinomial Coefficient."
From MathWorld https://mathworld.wolfram.com/MultinomialCoefficient.html
Subject classifications
distinct elements of multiplicity 
(
) is 
(Skiena 1990, p. 12).
is given by
is a binomial coefficient.
