Number of different values that obtain from a given set Suppose $S=\{1,2,\dots,7\}$, already we know that, $S$ have $2^7$ subsets. But the problem is, when we multiply elements of each subset $S$, how many different value we can obtain from multiplication of elements of each subsets? In $2^7$ subsets, some of them are equal, for example, for subset of size 2, $3\times 4=6\times 2$. Are there a simple way that eliminate the duplicate values for any subsets?

$7!=2^4\cdot 3^2\cdot 5^1\cdot 7^1$ has $(4+1)(2+1)(1+1)(1+1)=60$ divisors
Of these, the divisors who are multiples of $2^4$ but not multiples of $3$ are not possible to make (as $6$ would have been needed for that last factor of $2$ and would have brought a factor of $3$ along with it) as well as the divisors who are multiples of $9$ but not multiples of $2$ are not possible (for a similar reason).
There are $2\cdot 2 = 4$ factors who are multiples of $2^4$ but not multiples of $3$ (seen by looking at factors of the form $2^4\cdot 3^0\cdot 5^z\cdot 7^w$) and similarly there are $2\cdot 2 = 4$ factors who are multiples of $9$ but not multiples of $2$ (looking at $2^0\cdot 3^2\cdot 5^z\cdot 7^w$).
As such the total will be
$$60 - 4-4 = 52$$
