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Multinomial Theorem

From AoPSWiki

The Multinomial Theorem states that (a_1+a_2+\cdots+a_k)^n=\sum_{\substack{j_1,j_2,\ldots,j_k \\ 0 \leq j_i \leq n \textrm{ for each } i \\\textrm{and } j_1 + \l... where \binom{n}{j_1; j_2; \ldots ; j_k} is the multinomial coefficient \binom{n}{j_1; j_2; \ldots ; j_k}=\dfrac{n!}{j_1!\cdot j_2!\cdots j_k!}.

Note that this is a direct generalization of the Binomial Theorem: when k = 2 it simplifies to (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...

Contents

Proof

Using induction and the Binomial Theorem

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Combinatorial proof

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Problems

Introductory

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Intermediate

(x+y+z)^{2006}+(x-y-z)^{2006}

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

\mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad\math...

(Source: 2006 AMC 12A Problem 24)

Olympiad

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