How many numbers can you get by multiplying two or more distinct members of the set \(\{1,2,3,5,11\}\) together?
Use casework. First try finding the possiblities with only multiplying 2 numbers, and then the possibilities with multiplying 3 numbers, etc.
Using 2 numbers: \(2\cdot3, 2 \cdot 5, 2 \cdot 11, 3 \cdot 5, 3 \cdot 11, 5 \cdot 11.\)
Using 3 numbers: \(2 \cdot 3 \cdot 5, 2 \cdot 3 \cdot 11, 2 \cdot 5 \cdot 11, 3 \cdot 5 \cdot 11.\)
Using 4 numbers: \(2 \cdot 3 \cdot 5 \cdot 11.\)
So we have a total of \(6+4+4+1=\boxed{15}\)
