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Ken Ward's Mathematics Pages

Series Multinomial Theorem

Series Contents

Page Contents

  1. Multinomial Theorem
  2. Proof

Multinomial Theorem

The following is the multinomial formula or theorem, also called the Polynomial Theorem:
multinomialTheorem.gif [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:
multiNomialCoefficient.gif [1.2]

(Where the sum of the m's is equal to n).
For instance:
multiNomialCoefficientExample.gif [1.3]

Therefore, the Multinomial Theorem can be written:
multiNomialTheorem2.gif [1.4]


Proof

We can choose r0 items from n in: 
nChooser0.gif ways [1.5]
And similarly, we can choose any r in these ways:
multinomialTerms.gif [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:
multiNomialCoefficientTerm.gif [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:
multinomialTheorem.gif [1.1, repeated]







Ken Ward's Mathematics Pages


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