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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*b*c=2310? (The positive integers a, b, and c are distinct.)

 Jan 4, 2022
 #1
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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\((3^4-1)/2 =40\)

 Jan 4, 2022

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