# Multinomial Theorem

The **Multinomial Theorem** states that
where is the multinomial coefficient .

Note that this is a direct generalization of the Binomial Theorem: when it simplifies to

## Contents

## Proof

### Proof by Induction

Proving the Multinomial Theorem by Induction

For a positive integer and a non-negative integer ,

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:

### Combinatorial proof

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## Problems

### Intermediate

- The expression

is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

(Source: 2006 AMC 12A Problem 24)

### Olympiad

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