For a set of numbers \(T,\) we say that \(T\) has distinct subset sums if all distinct subsets of \(T\) have distinct sums. How many subsets of \(\{1,2,3,4,5,6,7,8\}\) have distinct subset sums?

**Details and assumptions**

The empty set (the set of no elements) has a sum of 0 by convention.

As an explicit example, the subset \( \{1, 2 \} \) satisfies the conditions, since it has \(2^2 = 4 \) subsets, whose sums are 0, 1, 2, and 3, which are distinct.

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