The following is the multinomial formula or theorem, also called the Polynomial Theorem: [1.1] While
it looks oppressive, it is easy to prove and also easy to use. In 1.1,
the a's are terms. n is an integer. The m's are the number of each term
selected. The multinomial coefficient can be written: [1.2]
(Where the sum of the m's is equal to n). For instance: [1.3]
Therefore, the Multinomial Theorem can be written: [1.4]
Proof
We can choose r0 items from n in: ways [1.5] And similarly, we can choose any r in these ways: [1.6]
That is, having chosen r0 items, we can choose r1 from the remaining n-r0, and similarly, for the other terms.
The coefficient of a term consisting of m0 a0's, m1 a1's, etc is the product of the above choices: [1.7]
Clearly, if we choose m0 of a0, then the power will be a0m0, and similarly, for the other a's. This leads to the Multinomial Theorem: [1.1, repeated]
[1.1]
[1.2]
[1.3]
[1.4]
ways [1.5]
[1.6]
[1.7]