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2003 AIME I Problems/Problem 3

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Problem

Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

Solution

Each element of the set will appear in $7$ two-element subsets, once with each other number.

$34$ will be the greater number in $7$ subsets.
$21$ will be the greater number in $6$ subsets.
$13$ will be the greater number in $5$ subsets.
$8$ will be the greater number in $4$ subsets.
$5$ will be the greater number in $3$ subsets.
$3$ will be the greater number in $2$ subsets.
$2$ will be the greater number in $1$ subsets.
$1$ will be the greater number in $0$ subsets.

Therefore the desired sum is $34\cdot7+21\cdot6+13\cdot5+8\cdot4+5\cdot3+3 \cdot2+2\cdot1+1\cdot0=\boxed{484}$ .

See also

2003 AIME I (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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Category: Introductory Combinatorics Problems

Looking for a challenging algebra text? Preparing for MATHCOUNTS or the AMC exams?
Check out Art of Problem Solving's Introduction to Algebra by Richard Rusczyk.

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