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Multinomial Series


A multinomial series is generalization of the binomial series discovered by Johann Bernoulli and Leibniz. The multinomial series arises in a generalization of the binomial distribution called the multinomial distribution.

It is given by

 (a_1+a_2+...+a_k)^n=sum_(n_1,n_2,...,n_k>=0; n_1+n_2+...+n_k=n)(n!)/(n_1!n_2!...n_k!)a_1^(n_1)a_2^(n_2)...a_k^(n_k),

where n=n_1+n_2+...+n_k.

For example,

 (a_1+a_2+a_3)^3=a_1^3+3a_1^2a_2+3a_1a_2^2+a_2^3+3a_1^2a_3+6a_1a_2a_3+3a_2^2a_3+3a_1a_3^2+3a_2a_3^2+a_3^3.

See also

Binomial Series, Multinomial Distribution

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Cite this as:

Weisstein, Eric W. "Multinomial Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultinomialSeries.html

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