Difference between revisions of "Multinomial Theorem"

Revision as of 11:12, 29 April 2008

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

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

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

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

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

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 is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
-<math> \mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514</math><math>\mathrm{(D) \ } 1,008,016\qquad\mathrm{(E) \ }  2,015,028</math>
+<math> \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</math>
 -([[2006_AMC_12A_Problems/Problem_24|2006 AMC 12A Problem 24]])
 ===Olympiad===