Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 8"
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== 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>. | + | The expression simplifies to <math>\left(\frac{x^{6}-1}{x-1}\right)^{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>. |
− | + | ---- | |
− | * [[University of South Carolina High School Math Contest/1993 Exam]] | + | |
+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 7|Previous Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 9|Next Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]] | ||
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 12:27, 23 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 .