How many distinct subsets of the set $S=\{1,8,9,39,52,91\}$ have three-digit sums?

34

{
 {{9, 91}},
 {{39, 91}},
 {{52, 91}},
 {{1, 8, 91}},
 {{1, 9, 91}},
 {{1, 39, 91}},
 {{1, 52, 91}},
 {{8, 9, 91}},
 {{8, 39, 91}},
 {{8, 52, 91}},
 {{9, 39, 52}},
 {{9, 39, 91}},
 {{9, 52, 91}},
 {{39, 52, 91}},
 {{1, 8, 9, 91}},
 {{1, 8, 39, 52}},
 {{1, 8, 39, 91}},
 {{1, 8, 52, 91}},
 {{1, 9, 39, 52}},
 {{1, 9, 39, 91}},
 {{1, 9, 52, 91}},
 {{1, 39, 52, 91}},
 {{8, 9, 39, 52}},
 {{8, 9, 39, 91}},
 {{8, 9, 52, 91}},
 {{8, 39, 52, 91}},
 {{9, 39, 52, 91}},
 {{1, 8, 9, 39, 52}},
 {{1, 8, 9, 39, 91}},
 {{1, 8, 9, 52, 91}},
 {{1, 8, 39, 52, 91}},
 {{1, 9, 39, 52, 91}},
 {{8, 9, 39, 52, 91}},
 {{1, 8, 9, 39, 52, 91}}
}