How many numbers can be expressed as the sum of two or more distinct elements of the set \(\{0, 1, 2, 4, 8, 16, 32, 64, 128\}\)?
The answer is 424.
This is equivalent to the question "How many numbers can be expressed with 8 or less digits in binary?"
Hence, every number from \(1_2\) to \(11{,}111{,}111_2\) works, for an answer of \(\boxed{255}\)
