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 r_{0} items from n in: ways [1.5] And similarly, we can choose any r in these ways: [1.6]

That is, having chosen r_{0} items, we can choose r_{1} from the remaining n-r_{0}, and similarly, for the other terms.

The coefficient of a term consisting of m_{0} a_{0}'s, m_{1} a_{1}'s, etc is the product of the above choices: [1.7]

Clearly, if we choose m_{0} of a_{0}, then the power will be a_{0}^{m0}, and similarly, for the other a's. This leads to the Multinomial Theorem: [1.1, repeated]

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