+364 For how many three-element sets (order doesn't matter) of positive integers {a,b,c} is it true that a*b*c=2310? (The positive integers a, b, and c are distinct.)
We can account for permutations by assuming WLOG that $a$ contains the prime factor 2. Thus, there are $3^4$ ways to position the other 4 prime numbers. Note that, with the exception of when all of the prime factors belong to $a$, we have over counted each case twice, as for when we put certain prime factors into $b$ and the rest into $c$, we count the exact same case when we put those prime factors which were in $b$ into $c$. Thus, our total number of cases is\((3^4-1)/2 =40\)
