Let the [set](https://artofproblemsolving.com/wiki/index.php/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.
Order the numbers in the set from greatest to least to reduce error: $\{34, 21, 13, 8, 5, 3, 2, 1\}.$
Each [element](https://artofproblemsolving.com/wiki/index.php/Element) of the [set](https://artofproblemsolving.com/wiki/index.php/Set) will appear in $7$ two-element [subsets](https://artofproblemsolving.com/wiki/index.php/Subset), 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 $34\cdot7+21\cdot6+13\cdot5+8\cdot4+5\cdot3+3 \cdot2+2\cdot1+1\cdot0=\boxed{484}$.
Note: Note that $7+6+5+4+3+2+1=\binom{8}{2}$, so we have counted all the possible cases.
~Yiyj1