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Concept
Multinomial theorem
Summary
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 from binomials to multinomials.
For any positive integer m and any non-negative integer n, the multinomial formula describes 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.
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. 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
The statement of the theorem can be written concisely using multiindices:
where
and
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:
The numbers
appearing in the theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials:
The substitution of x_i = 1 for all i into the multinomial theorem
gives immediately that
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, .
Official source
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