Difference between revisions of "2008 AMC 12A Problems/Problem 19"
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In the expansion of | In the expansion of | ||
− | < | + | <cmath>\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,</cmath> |
− | what is the coefficient of <math>x^{28}</math>? | + | what is the [[coefficient]] of <math>x^{28}</math>? |
<math>\textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405</math> | <math>\textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405</math> | ||
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==See Also== | ==See Also== | ||
{{AMC12 box|year=2008|ab=A|num-b=18|num-a=20}} | {{AMC12 box|year=2008|ab=A|num-b=18|num-a=20}} | ||
+ | |||
+ | [[Category:Introductory Algebra Problems]] |
Revision as of 18:01, 20 February 2008
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
. 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 |