Multinomial theorem#Multinomial coefficients

The multinomial or Polynomialkoeffizient is an extension of the binomial coefficient. For non-negative integers, and it is defined as

The faculty is from.

• 2.3.2.1 example

Properties

The multinomial are always integers.

The multinomial can be expressed with the binomial coefficients as

Applications and interpretations

Multinomial

In generalization of the binomial theorem called Multinomialtheorem applies (also Polynomialsatz )

From the multinomial theorem immediately follows:

Multinomial

Application find those coefficients in the multinomial

A probability distribution of discrete random variables.

Combinatorial interpretations

Objects in boxes

The multinomial coefficient is the number of ways to create objects in boxes, in which objects are exactly in the first box into the second box objects, etc.

Example

How many different ways are there to place 32 cards of Skat of 10 cards to the players as well as 3 to 2 remaining tickets to the " skat "?

Since there are objects that are divided into boxes, where objects are in the first three boxes depending on objects and in the fourth case, the number of possibilities is given by the multinomial:

Arrangement of things

The multinomial coefficient also indicates the number of different arrangements of objects, wherein the first - time ( indistinguishable ) can occur, the second - time, etc.

Example

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

So Wanted is the number of ways to arrange 11 things, the first ("M" ) times, the second ("I" ) times ( indistinguishable ) occurs, and the third ("S" ) as well, and the fourth ( " P ") times. So this is the multinomial coefficient

By comparison, the number of different ways to organize eleven completely different things in rows, with 11! = 39.9168 million much higher.

Pascal's simplices

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