Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 8"
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== Problem == | == Problem == | ||
| + | What is the coefficient of <math>x^3</math> in the expansion of | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } | + | <center><math>4 (1 + x + x^2 + x^3 + x^4 + x^5 )^6? </math></center> |
| + | |||
| + | <center><math> \mathrm{(A) \ } 40 \qquad \mathrm{(B) \ }48 \qquad \mathrm{(C) \ }56 \qquad \mathrm{(D) \ }62 \qquad \mathrm{(E) \ } 64 </math></center> | ||
== Solution == | == Solution == | ||
| + | The expression simplifies to <math>(\frac{x^{6}-1}{x-1})^{6}</math>. Expanding both the numerator and denominator, we see that the coefficient of the <math>x^{3}</math> term is <math>{6\choose 5}+{6\choose 3}+{6\choose 6}+{6\choose 3}=56</math>. | ||
== See also == | == See also == | ||
* [[University of South Carolina High School Math Contest/1993 Exam]] | * [[University of South Carolina High School Math Contest/1993 Exam]] | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
| + | [[Category:Introductory Algebra Problems]] | ||
Revision as of 20:40, 22 July 2006
Problem
What is the coefficient of 
in the expansion of
Solution
The expression simplifies to 
. Expanding both the numerator and denominator, we see that the coefficient of the 
term is 
.
