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What is the multinomial coefficient?

The multinomial theorem provides an easy way to expand the power of a sum of variables. As “multinomial” is just another word for polynomial, this could also be called the *polynomial *theorem.

It tells us that when you expand any multinomial (x_{1}+ x_{2} + ….x_{k})^{n} the coefficients of every term x_{1}^{n1} x_{2}^{n2}….x_{k}^{nk} in the resulting polynomial will be:

This is called the multinomial coefficient, and n_{1}, n_{2},…., n_{k} are integers which add up to make *n*.

So

The exclamation mark signifies a factorial, where you multiply an integer by all the integers smaller than it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

The binomial theorem is a special case of the multinomial theorem.

## The Multinomial Theorem in Combinatorics

Suppose you have n distinct, differentiable items you are placing in k distinct groups. If you place n_{1} item group 1, n_{2} items in group two, and so on till you place n_{k} items in the last group, the number of distinguishable permutations is given by the multinomial coefficient:

The multinomial coefficient can also be used to find the number of distinguishable permutations of n objects when n= n_{1} + n _{2} +….+n_{k} and you have n_{1} items of kind 1, n_{2} items of type 2, and n_{k} items of type k for every k. Here again the number of permutations is given by the multinomial formula (above).

## Examples

How many words can you make from the letters in *mathematical*? The word mathematical contains 2 m, 3 a, 2 t, one h, one e, one i, one c, and one l. Use the formula above, and you find the the number of permutations will be

. Work the factorial, and you get 19958400. So there are 19958400 possible permutations of the letters in the word *mathematical*.

For another example, divide a class of 15 into groups of 3 for a final project. The groups are numbered a through e. There are

ways of divying up the class. Group the students randomly, and any student will have a 1 in 168168000 chance of ending up in any given group with any particular two friends.

## Sources

Introduction to Probability: Multinomial Coefficients

Permutations and Combinations

Multinomial Coefficients