Multinomial coefficient

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The multinomial coefficient or polynomial coefficient is an extension of the binomial coefficient . For nonnegative integers and it is defined as

Where is the faculty of .


The multinomial coefficients are always whole numbers.

The multinomial coefficients can also be expressed with the binomial coefficients as


Applications and interpretations

Multinomial theorem

In generalizing the binomial theorem , the so-called multinomial theorem (also polynomial theorem ) applies


From the multinomial theorem it follows immediately:

Multinomial distribution

Those coefficients are also used in the multinomial distribution


a probability distribution of discrete random variables.

Combinatorial Interpretations

Objects in boxes

The multinomial coefficient indicates the number of possibilities for placing objects in boxes, with objects in the first box, objects in the second box , and so on.


How many different ways there are of the 32 cards of Skat to give each 10 cards to three players and two cards in the "Skat" when the order of the cards is not respected?

Since these are objects that are to be divided into boxes, with objects in the first three boxes and objects in the fourth box , the number of possibilities is given by the following multinomial coefficients:

Arrangement of things

The multinomial coefficient also indicates the number of different arrangements of things, where the first occurs (indistinguishable), the second occurs, etc.


How many different "words" can be formed from the letters MISSISSIPPI?

We are looking for the number of possibilities to arrange 11 things, whereby the first ("M") -time, the second ("I") -time (indistinguishable) occurs, the third ("S") also and the fourth (" P ") times. So this is the multinomial coefficient

For comparison: the number of possibilities to arrange eleven completely different things in rows is 11! = 39,916,800 much higher.

Pascal's Simplizes

Analogous to the Pascal triangle of the binomial coefficients, the -th multinomial coefficients can also be arranged as geometric figures ( simplices ): The trinomial coefficients lead to the Pascal pyramid , the other to -dimensional Pascal simplices .

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