 TOPICS  # 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).

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/

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## 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