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
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.
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. http://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 Coefficient
Cite this as:
Weisstein, Eric W. "Multinomial Coefficient."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/MultinomialCoefficient.html
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