For how many three-element sets (order doesn't matter) of positive integers $\{a,b,c\}$ is it true that $a\times b\times c=2310$?  (The positive integers $a,$ $b,$ and $c$ are distinct.)
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
For how many three-element sets (order doesn't matter) of positive integers $\{a,b,c\}$ is it true that $a\times b\times c=2310$?  (The positive integers $a,$ $b,$ and $c$ are distinct.)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
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For how many three-element sets (order doesn't matter) of positive integers $\{a,b,c\}$ is it true that $a\times b\times c=2310$?  (The positive integers $a,$ $b,$ and $c$ are distinct.)
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