Difference between revisions of "2006 AMC 12A Problems/Problem 24"

(Solution)
(See also)
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== See also ==
 
== See also ==
 
* [[2006 AMC 12A Problems]]
 
* [[2006 AMC 12A Problems]]
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by the multi-nomial theorem, the expressions of the two are:
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 +
<math>\sum{\frac{2006!}{a!b!c!}x^ay^bz^c}</math>
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and:
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<math>\sum{\frac{2006!}{a!b!c!}x^a(-y)^b(-z)^c}</math>
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 +
respectively, where <math>a+b+c = 2006</math>. Since the coefficients of like terms are the same in each expression, each like term either cancel one another out or the coefficient doubles. In each expansion there are:
 +
 +
<math>{2006+2\choose 2} = 2015028</math>
 +
 +
terms without cancellation. For any term in the second expansion to be negative, the parity of the exponents of <math>y</math> and <math>z</math> must be opposite. Now we find a pattern:
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 +
if the exponent of <math>y</math> is 1, the exponent of <math>z</math> can be all even integers up to 2004, so 1003 terms.
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if the exponent of <math>y</math> is 3, the exponent of <math>z</math> can go up to 2002, so 1002 terms.
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<math>\vdots</math>
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if the exponent of <math>y</math> is 2005$, then <math>z</math> can only be 0. So 1 term.
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add them up we get <math>\frac{1003*1004}{2}</math> terms. However, we can switch the exponents of <math>y</math> and <math>z</math> and these terms will still have a negative sign. So there are a total of <math>1003*1004</math> negative terms.
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Subtract this number from 2015028 we obtain <math>D. 1008016</math> as our answer.

Revision as of 13:06, 12 July 2006

Problem

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$$\mathrm{(D) \ } 1,008,016\qquad\mathrm{(E) \ }  2,015,028$

Solution

by the multi-nomial theorem, the expressions of the two are:

$\sum{\frac{2006!}{a!b!c!}x^ay^bz^c}$

and:

$\sum{\frac{2006!}{a!b!c!}x^a(-y)^b(-z)^c}$

respectively, where $a+b+c = 2006$. Since the coefficients of like terms are the same in each expression, each like term either cancel one another out or the coefficient doubles. In each expansion there are:

$\binom{2006+2}{2} = 2015028$

terms without cancellation. For any term in the second expansion to be negative, the parity of the exponents of $y$ and $z$ must be opposite. Now we find a pattern:

if the exponent of $y$ is 1, the exponent of $z$ can be all even integers up to $2004$, so 1003 solutions.

if the exponent of $y$ is 3, the exponent of $z$ can go up to $2002$, so 1002 solutions.

$\vdots$

if the exponent of $y$ is 2005$, then $z$ can only be $0$. So 1 solution.

add them up we get $\frac{1003*1004}{2}$ solutions. However, we can switch the exponents of $y$ and $z$ and these terms will still have a negative sign. So there are a total of $1003*1004$ negative terms.

Subtract this number from 2015028 we obtain $D. 1008016$ as our answer.

See also

by the multi-nomial theorem, the expressions of the two are:

$\sum{\frac{2006!}{a!b!c!}x^ay^bz^c}$

and:

$\sum{\frac{2006!}{a!b!c!}x^a(-y)^b(-z)^c}$

respectively, where $a+b+c = 2006$. Since the coefficients of like terms are the same in each expression, each like term either cancel one another out or the coefficient doubles. In each expansion there are:

${2006+2\choose 2} = 2015028$

terms without cancellation. For any term in the second expansion to be negative, the parity of the exponents of $y$ and $z$ must be opposite. Now we find a pattern:

if the exponent of $y$ is 1, the exponent of $z$ can be all even integers up to 2004, so 1003 terms.

if the exponent of $y$ is 3, the exponent of $z$ can go up to 2002, so 1002 terms.

$\vdots$

if the exponent of $y$ is 2005$, then $z$ can only be 0. So 1 term.

add them up we get $\frac{1003*1004}{2}$ terms. However, we can switch the exponents of $y$ and $z$ and these terms will still have a negative sign. So there are a total of $1003*1004$ negative terms.

Subtract this number from 2015028 we obtain $D. 1008016$ as our answer.