# Difference between revisions of "Multinomial Theorem"

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where <math>\binom{n}{j_1; j_2; \ldots ; j_k}</math> is the [[multinomial coefficient]] <math>\binom{n}{j_1; j_2; \ldots ; j_k}=\dfrac{n!}{j_1!\cdot j_2!\cdots j_k!}</math>. | where <math>\binom{n}{j_1; j_2; \ldots ; j_k}</math> is the [[multinomial coefficient]] <math>\binom{n}{j_1; j_2; \ldots ; j_k}=\dfrac{n!}{j_1!\cdot j_2!\cdots j_k!}</math>. | ||

− | Note that this is a direct generalization of the [[Binomial Theorem]] | + | Note that this is a direct generalization of the [[Binomial Theorem]], when <math>k = 2</math> it simplifies to |

<cmath> | <cmath> | ||

(a_1 + a_2)^n = \sum_{\substack{0\leq j_1, j_2 \leq n \\ j_1 + j_2 = n}} \binom{n}{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j} | (a_1 + a_2)^n = \sum_{\substack{0\leq j_1, j_2 \leq n \\ j_1 + j_2 = n}} \binom{n}{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j} | ||

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=== Combinatorial proof === | === Combinatorial proof === | ||

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{{stub}} | {{stub}} | ||

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===Olympiad=== | ===Olympiad=== | ||

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[[Category:Theorems]] | [[Category:Theorems]] | ||

[[Category:Combinatorics]] | [[Category:Combinatorics]] | ||

[[Category:Multinomial Theorem]] | [[Category:Multinomial Theorem]] |

## Latest revision as of 19:37, 4 January 2023

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

*This article is a stub. Help us out by expanding it.*

## 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)