The multinomial theorem describes how to expand the power of a sum of more than two terms. It is a generalization of the binomial theorem to polynomials with any number of terms. It expresses a power as a weighted sum of monomials of the form where the weights are given by generalizations of binomial coefficients called multinomial coefficients.
The first important definition is the multinomial coefficient:
For non-negative integers such that the multinomial coefficient is
Note that when this is the binomial coefficient:
Now the multinomial theorem can be stated as follows:
For a positive integer and a non-negative integer ,
The number of terms of this sum are given by a stars and bars argument: it is .
There are two proofs of the multinomial theorem, an algebraic proof by induction and a combinatorial proof by counting. The algebraic proof is presented first.
Proceed by induction on
When the result is true, and when the result is the binomial theorem. Assume that and that the result is true for When
Treating as a single term and using the induction hypothesis,
By the binomial theorem, this becomes
Since this can be rewritten as
Here is the combinatorial proof, which relies on a fact from the Multinomial Coefficients wiki.
Consider a term in the expansion of . It must be of the form for some integer and non-negative integers Since each term must come from choosing one summand from we must have The number of different ways that we can get this term will be the number of ways to choose copies of , copies of , and so on. By a result from the Multinomial Coefficients wiki, this is exactly
How many terms are in the expansion of
There is one term for each ordered triple with . One way to count these triples is to represent them as collections of bars and stars; for instance,
represents the triple . The number of such collections is , so there are terms in the expansion.
Determine the coefficient of in the expansion of the polynomial
A general term in the expansion of will be of the form To have the term with we need This gives us
implying the answer is 590625.
Find the coefficient of in the expansion of
A general term of the expansion has the form In order to have a coefficient of we must have and We can write and
Thus, the coefficient will be the sum such that and
We can evaluate this as