Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 20"
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== Problem == | == Problem == | ||
| + | Let <math>A_1, A_2, \ldots , A_{63}</math> be the 63 nonempty subsets of <math>\{ 1,2,3,4,5,6 \}</math>. For each of these sets <math>A_i</math>, let <math>\pi(A_i)</math> denote the product of all the elements in <math>A_i</math>. Then what is the value of <math>\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})</math>? | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center><math> \mathrm{(A) \ }5003 \qquad \mathrm{(B) \ }5012 \qquad \mathrm{(C) \ }5039 \qquad \mathrm{(D) \ }5057 \qquad \mathrm{(E) \ }5093 </math></center> |
== Solution == | == Solution == | ||
| + | We have <math>(1+1)(1+2)(1+3)(1+4)(1+5)(1+6)-1</math> (The <math>-1</math> since we have one less set). This is <math>7!-1=5039</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]] | ||
Revision as of 20:10, 22 July 2006
Problem
Let 
be the 63 nonempty subsets of 
. For each of these sets 
, let 
denote the product of all the elements in . Then what is the value of 
?
Solution
We have 
(The 
since we have one less set). This is 
.
