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Multinomial Coefficient
The multinomial coefficients
 |
(1)
|
are the terms in the multinomial series expansion. In other words, the number of distinct permutations in a multiset
of 
distinct elements of multiplicity 
(
) is 
(Skiena 1990, p. 12).
The multinomial coefficient is returned by the Wolfram Language function Multinomial[n1, n2, ...].
The special case 
is given by
 |
(2)
|
where 
is a binomial coefficient.
The multinomial coefficients satisfy
 |  |  |
(3)
|
 |  |  |
(4)
|
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 TheoremRelated Wolfram sites
https://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. New York: Dover, pp. 823-824, 1972.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. https://www.inwap.com/pdp10/hbaker/hakmem/number.html#item42.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 113, 1992.Referenced on Wolfram|Alpha
Multinomial CoefficientCite this as:
Weisstein, Eric W. "Multinomial Coefficient." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultinomialCoefficient.html