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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
of the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
from binomials to multinomials.


Theorem

For any positive integer and any non-negative integer , the multinomial theorem describes how a sum with terms expands when raised to the th power: (x_1 + x_2 + \cdots + x_m)^n = \sum_ x_1^ \cdot x_2^ \cdots x_m^ where = \frac is a multinomial coefficient. The sum is taken over all combinations of
nonnegative In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
indices through such that the sum of all is . That is, for each term in the expansion, the exponents of the must add up to . In the case , this statement reduces to that of the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
.


Example

The third power of the trinomial is given by (a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c + 3 c^2 a + 3 c^2 b + 6 a b c. This can be computed by hand using the
distributive property In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary ...
of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example, the term a^2 b^0 c^1 has coefficient = \frac = \frac = 3, the term a^1 b^1 c^1 has coefficient = \frac = \frac = 6, and so on.


Alternate expression

The statement of the theorem can be written concisely using multiindices: :(x_1+\cdots+x_m)^n = \sum_x^\alpha where : \alpha=(\alpha_1,\alpha_2,\dots,\alpha_m) and : x^\alpha=x_1^ x_2^ \cdots x_m^


Proof

This proof of the multinomial theorem uses the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
and induction on . First, for , both sides equal since there is only one term in the sum. For the induction step, suppose the multinomial theorem holds for . Then : \begin & (x_1+x_2+\cdots+x_m+x_)^n = (x_1+x_2+\cdots+(x_m+x_))^n \\ pt= & \sum_ x_1^ x_2^\cdots x_^(x_m+x_)^K \end by the induction hypothesis. Applying the binomial theorem to the last factor, : = \sum_ x_1^x_2^\cdots x_^\sum_x_m^x_^ : = \sum_ x_1^x_2^\cdots x_^x_m^x_^ which completes the induction. The last step follows because : = , as can easily be seen by writing the three coefficients using factorials as follows: : \frac \frac=\frac.


Multinomial coefficients

The numbers : appearing in the theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s or of
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
s: : = \frac = \cdots


Sum of all multinomial coefficients

The substitution of for all into the multinomial theorem :\sum_ x_1^ x_2^ \cdots x_m^ = (x_1 + x_2 + \cdots + x_m)^n gives immediately that : \sum_ = m^n.


Number of multinomial coefficients

The number of terms in a multinomial sum, , is equal to the number of monomials of degree on the variables : : \#_ = . The count can be performed easily using the method of stars and bars.


Valuation of multinomial coefficients

The largest power of a prime that divides a multinomial coefficient may be computed using a generalization of Kummer's theorem.


Asymptotics

By
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
, or equivalently the log-gamma function's asymptotic expansion, \log\binom = k n \log(k) + \frac \left(\log(k) - (k - 1) \log(2 \pi n)\right) - \frac + \frac - \frac + O\left(\frac\right)so for example,\binom \sim \frac


Interpretations


Ways to put objects into bins

The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing distinct objects into distinct bins, with objects in the first bin, objects in the second bin, and so on.


Number of ways to select according to a distribution

In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
and
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, if one has a number distribution of labels, then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution on a set of total items, represents the number of items to be given the label . (In statistical mechanics is the label of the energy state.) The number of arrangements is found by *Choosing of the total to be labeled 1. This can be done \tbinom ways. *From the remaining items choose to label 2. This can be done \tbinom ways. *From the remaining items choose to label 3. Again, this can be done \tbinom ways. Multiplying the number of choices at each step results in: :\cdots=\frac \cdot \frac \cdot \frac\cdots. Cancellation results in the formula given above.


Number of unique permutations of words

The multinomial coefficient :\binom is also the number of distinct ways to permute a
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the ''multiplicity'' of ...
of elements, where is the multiplicity of each of the th element. 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 : = \frac = 34650.


Generalized Pascal's triangle

One can use the multinomial theorem to generalize
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup table for multinomial coefficients.


See also

*
Multinomial distribution In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided die rolled ''n'' times. For ''n'' statistical independence, indepen ...
*
Stars and bars (combinatorics) In combinatorics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems. It can be used to solve a variety of counting problems, such as how many ...


References

{{Reflist Combinatorics Factorial and binomial topics Articles containing proofs Theorems about polynomials