Multinomial Coefficient

The multinomial coefficients


are the terms in the multinomial series expansion. In other words, the number of distinct permutations in a multiset of k distinct elements of multiplicity n_i (1<=i<=k) is (n_1,...,n_k)! (Skiena 1990, p. 12).

The multinomial coefficient is returned by the Wolfram Language function Multinomial[n1, n2, ...].

The special case (m,n)! is given by

 (m,n)!=(m+n; m)=(m+n; n)=((m+n)!)/(m!n!),

where (n; k) 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

Explore with Wolfram|Alpha


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., 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.

Subject classifications