Difference between revisions of "2008 AMC 12A Problems/Problem 19"
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==Problem== | ==Problem== | ||
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> | <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>\ | + | <math>\mathrm{(A)}\ 195\qquad\mathrm{(B)}\ 196\qquad\mathrm{(C)}\ 224\qquad\mathrm{(D)}\ 378\qquad\mathrm{(E)}\ 405</math> |
==Solution== | ==Solution== | ||
Revision as of 00:48, 26 April 2008
Problem
In the expansion of
![\[\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,\]](http://latex.artofproblemsolving.com/e/4/2/e42212799afc8b50d066ff0471255746cf718dec.png)
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 | |
