Multinomial theorem
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials.
Theorem
For any positive integer m and any nonnegative integer n, the multinomial formula tells us how a sum with m terms expands when raised to an arbitrary power n:
where
is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices k_{1} through k_{m} such that the sum of all k_{i} is n. That is, for each term in the expansion, the exponents of the x_{i} must add up to n. Also, as with the binomial theorem, quantities of the form x^{0} that appear are taken to equal 1 (even when x equals zero).
In the case m = 2, this statement reduces to that of the binomial theorem.
Example
The third power of the trinomial a + b + c is given by
This can be computed by hand using the distributive property of multiplication over addition, but it can also be done (perhaps more easily) with the multinomial theorem, which gives us a simple formula for any coefficient we might want. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example:
 has the coefficient
 has the coefficient .
Alternate expression
The statement of the theorem can be written concisely using multiindices:
where α = (α_{1},α_{2},…,α_{m}) and x^{α} = x_{1}^{α1}x_{2}^{α2}⋯x_{m}^{αm}.
Proof
This proof of the multinomial theorem uses the binomial theorem and induction on m.
First, for m = 1, both sides equal x_{1}^{n} since there is only one term k_{1} = n in the sum. For the induction step, suppose the multinomial theorem holds for m. Then
by the induction hypothesis. Applying the binomial theorem to the last factor,
which completes the induction. The last step follows because
as can easily be seen by writing the three coefficients using factorials as follows:
Multinomial coefficients
The numbers
which can also be written as
are the multinomial coefficients. Just like "n choose k" are the coefficients when a binomial is raised to the n^{th} power (e.g., the coefficients are 1,3,3,1 for (a + b)^{3}, where n = 3), the multinomial coefficients appear when a multinomial is raised to the n^{th} power (e.g., (a + b + c)^{3}).
Sum of all multinomial coefficients
The substitution of x_{i} = 1 for all i into:
gives immediately that
Number of multinomial coefficients
The number of terms in a multinomial sum, #_{n,m}, is equal to the number of monomials of degree n on the variables x_{1}, …, x_{m}:
The count can be performed easily using the method of stars and bars.
Central multinomial coefficients
All of the multinomial coefficients for which the following holds true:
are central multinomial coefficients: the greatest ones and all of equal size.
A special case for m = 2 is the central binomial coefficient.
Interpretations
Ways to put objects into boxes
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k_{1} objects in the first bin, k_{2} objects in the second bin, and so on.^{[1]}
Number of ways to select according to a distribution
In statistical mechanics and combinatorics if one has a number distribution of labels then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution {n_{i}} on a set of N total items, n_{i} represents the number of items to be given the label i. (In statistical mechanics i is the label of the energy state.)
The number of arrangements is found by
 Choosing n_{1} of the total N to be labeled 1. This can be done ways.
 From the remaining N − n_{1} items choose n_{2} to label 2. This can be done ways.
 From the remaining N − n_{1} − n_{2} items choose n_{3} to label 3. Again, this can be done ways.
Multiplying the number of choices at each step results in:
Upon cancellation, we arrive at the formula given in the introduction.
Number of unique permutations of words
The multinomial coefficient is also the number of distinct ways to permute a multiset of n elements, and k_{i} are the multiplicities of each of the distinct elements. For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps is
(This is just like saying that there are 11! ways to permute the letters—the common interpretation of factorial as the number of unique permutations. However, we created duplicate permutations, because some letters are the same, and must divide to correct our answer.)
Generalized Pascal's triangle
One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup table for multinomial coefficients.
The case of n = 3 can be easily drawn by hand. The case of n = 4 can be drawn with effort as a series of growing pyramids.
See also
References
 ↑ National Institute of Standards and Technology (May 11, 2010). "NIST Digital Library of Mathematical Functions". Section 26.4. Retrieved August 30, 2010.
External links

mutinom.m
function in Specfun (since 1.1.0) package of OctaveForge for GNU Octave. SVN version  Hazewinkel, Michiel, ed. (2001), "Multinomial coefficient", Encyclopedia of Mathematics, Springer, ISBN 9781556080104