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Multinomial Theorem

Revision as of 10:12, 29 April 2008 by 1=2 (talk | contribs) (→‎Intermediate)

The multinomial theorem states that

$(a_1+a_2+\cdots+a_x)^n=\sum_{k_1,k_2,\cdots,k_x}\binom{n}{k_1,k_2,\cdots,k_x}a_1^{k_1}a_2^{k_2}\cdots a_x^{k_x}$

where

$\binom{n}{k_1,k_2,\cdots,k_x}=\dfrac{n!}{k_1!k_2!\cdots k_x!}$

Problems

Introductory

Intermediate

  • The expression

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

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

$\mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad\mathrm{(E) \ } 2,015,028$

-(2006 AMC 12A Problem 24)

Olympiad

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