Multinomial Coefficient Calculator

Instructions: Use this Multinomial Coefficient Calculator to compute showing all the steps a multinomial coefficient \(\displaystyle {N \choose k_1 k_2 ... k_j}\), using the form below:

N =
k values (comma or space separated. Integers and must add up to \(N\)) =

The Multinomial Coefficients

The multinomial coefficient is widely used in Statistics, for example when computing probabilities with the hypergeometric distribution.

By definition, the hypergeometric coefficients are defined as:

\[ \displaystyle {N \choose k_1 k_2 ... k_j} = \frac{N!}{k_1! k_2! ... k_j!} \]

with \(k_1 + k_2 + ... + k_j = N\). By observing at the form above, the multinomial coefficient is clearly a generalization of the combinatorial coefficient, only that instead of two combinations, you have \(j\) combinations.

Other Applications

The multinomial coefficients are also useful for a multiple sum expansion that generalizes the Binomial Theorem, but instead of summing two values, we sum \(j\) values.

Question for you: Do you think that there is something similar as the Pascal Triangle for multinomial coefficients as there is for binomial coefficients?

In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us.

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