<s>[INST] How many numbers can you get by multiplying two or more distinct members of the set $\{1,2,3,5,11\}$ together? Apparently the answer is 15,  but I got 26.
My process:
The set $\{1,2,3,5,11\}$ has five numbers.
Ways to choose two members: $_5C_2$
Ways to choose three members: $_5C_3$
Ways to choose four members: $_5C_4$
Ways to choose five members: $_5C_5$
Total: $_5C_2 + {}_5C_3 + {}_5C_4 + {}_5C_5 = 26$
Is there a detail I have misunderstood in this question?
 [/INST] Note that, because two or more members can be multiplied, multiplying by $1$ will only make a difference if it is one of two numbers. Thus, multiplying by $1$ adds four potential numbers.
Now, we only need to consider the number of combinations that can be made from $2$, $3$, $5$, and $11$.
Choosing two from this set offers six possiblities: $2 \cdot 3$, $2 \cdot 5$, $2 \cdot 11$, $3 \cdot 5$, $3 \cdot 11$, and $5 \cdot 11$.
Choosing three offers four possibilities: $2 \cdot 3 \cdot 5$, $2 \cdot 3 \cdot 11$, $2 \cdot 5 \cdot 11$, and $3 \cdot 5 \cdot 11$.
Finally, there is one possibility with four chosen: $2 \cdot 3 \cdot 5 \cdot 11$. Thus, there are $4 + 6 + 4 + 1 = \boxed{15}$.

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