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University of South Carolina High School Math Contest/1993 Exam/Problem 20

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Problem

Let $A_1, A_2, \ldots , A_{63}$ be the 63 nonempty subsets of $\{ 1,2,3,4,5,6 \}$ . For each of these sets $A_i$ , let $\pi(A_i)$ denote the product of all the elements in $A_i$ . Then what is the value of $\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})$ ?

$\mathrm{(A) \ }5003 \qquad \mathrm{(B) \ }5012 \qquad \mathrm{(C) \ }5039 \qquad \mathrm{(D) \ }5057 \qquad \mathrm{(E) \ }50...$

Solution

We have $(1+1)(1+2)(1+3)(1+4)(1+5)(1+6)-1$ (The $-1$ since we have one less set). This is $7!-1=5039$ .

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