Difference between revisions of "2008 AMC 12A Problems/Problem 19"
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Let <math>A = \left(1 + x + x^2 + \cdots + x^{14}\right)</math> and <math>B = \left(1 + x + x^2 + \cdots + x^{27}\right)</math>. We are expanding <math>A \cdot A \cdot B</math>. | Let <math>A = \left(1 + x + x^2 + \cdots + x^{14}\right)</math> and <math>B = \left(1 + x + x^2 + \cdots + x^{27}\right)</math>. We are expanding <math>A \cdot A \cdot B</math>. | ||
− | Since there are <math>15</math> terms in <math>A</math>, there are <math>15^2 = 225</math> ways to choose one term from each <math>A</math>. The product of the selected terms is <math>x^n</math> for some integer <math>n</math> between <math>0</math> and <math>28</math> inclusive. For each <math>n \neq 0</math>, there is one and only one <math>x^{28 - n}</math> in <math>B</math>. For example, if I choose <math>x^2</math> from <math>A</math> , then there is exactly one power of x in <math>B</math> that I can choose; in this case, it would be <math>x^24</math>. Since there is only one way to choose one term from each <math>A</math> to get a product of <math>x^0</math>, there are <math>225 - 1 = 224</math> ways to choose one term from each <math>A</math> and one term from <math>B</math> to get a product of <math>x^{28}</math>. Thus the coefficient of the <math>x^{28}</math> term is <math>224 \Rightarrow C</math>. | + | Since there are <math>15</math> terms in <math>A</math>, there are <math>15^2 = 225</math> ways to choose one term from each <math>A</math>. The product of the selected terms is <math>x^n</math> for some integer <math>n</math> between <math>0</math> and <math>28</math> inclusive. For each <math>n \neq 0</math>, there is one and only one <math>x^{28 - n}</math> in <math>B</math>. For example, if I choose <math>x^2</math> from <math>A</math> , then there is exactly one power of <math>x</math> in <math>B</math> that I can choose; in this case, it would be <math>x^(24)</math>. Since there is only one way to choose one term from each <math>A</math> to get a product of <math>x^0</math>, there are <math>225 - 1 = 224</math> ways to choose one term from each <math>A</math> and one term from <math>B</math> to get a product of <math>x^{28}</math>. Thus the coefficient of the <math>x^{28}</math> term is <math>224 \Rightarrow C</math>. |
==See Also== | ==See Also== |
Revision as of 23:50, 22 November 2011
Problem
In the expansion of
what is the coefficient of
?
Solution
Let and
. We are expanding
.
Since there are terms in
, there are
ways to choose one term from each
. The product of the selected terms is
for some integer
between
and
inclusive. For each
, there is one and only one
in
. For example, if I choose
from
, then there is exactly one power of
in
that I can choose; in this case, it would be
. Since there is only one way to choose one term from each
to get a product of
, there are
ways to choose one term from each
and one term from
to get a product of
. Thus the coefficient of the
term is
.
See Also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |