Multinomial theorem

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the multinomial theorem says 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

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

 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

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 indices k₁ 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

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

, quantities of the form x⁰ 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 = (α₁,α₂,…,α_m) and x^α = x₁^α₁x₂^α₂⋯x_m^α_m.

Proof

This proof of the multinomial theorem uses the binomial theorem

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

 and induction

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

 on m.

First, for m = 1, both sides equal x₁ⁿ since there is only one term k₁ = 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 you raise a binomial 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 one raises a multinomial 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 multinomial sum, #_n,m, is equal to the number of monomials of degree n on the variables x₁, …, x_m:

The count can be easily performed 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 central binomial coefficient

Central binomial coefficient

In mathematics the nth central binomial coefficient is defined in terms of the binomial coefficient byThey are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle...

.

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 k₁ objects in the first bin, k₂ objects in the second bin, and so on.

Number of ways to select according to a distribution

In statistical mechanics

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

 and combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 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 n₁ of the total N to be labeled 1. This can be done ways.
• From the remaining N − n₁ items choose n₂ to label 2. This can be done ways.
• From the remaining N − n₁ − n₂ items choose n₃ 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

In addition, the multinomial coefficient is also the number of distinct ways to permute

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

 a multiset

Multiset

In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...

 of n elements, and k_i are the multiplicities

Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....

 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

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 as the number of unique permutations. However, we created duplicate permutations, due to the fact that 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

Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

 or Pascal's pyramid

Pascal's pyramid

In mathematics, Pascal's Pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's Pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial...

 to Pascal's simplex

Pascal's simplex

In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.- Induction of Pascal's simplices :...

. 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.