sets with sum of 3 digit

Question

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

I know we have $64$ total subsets, but I don't think that helps us in this question ... does it?

Answer 1

The rationale behind the selection is ${91,52,39,9,(8,1)} = {a1,a2,a3,a4,a5}$ The way I have grouped from the second element to the last element (i.e from a2 to a5) when added alone or a combination thereof will have a distinct set with sum >=100. The number of ways are $4\left({3\choose1}+{3\choose2}+{3\choose3}\right)+1 = 29$. Further${(52,39),9,(8,1)} = {b1,b2,b3}$. The way I have grouped the second element to the last element(i.e from b2 to b3) when added alone or a combination thereof will have a distinct set with sum >=100. The number of such sets is $2\left({2\choose1}\right)+1=5$. Thus the total number of distinct sets where the sum of their elements is a three digit = $34$