How many ways to generate unique multiplication result from given set? From set {2, 2, 3, 5}, I can have 8 ways to generate unique multiplication result, which:
- two number multiplication: 2*2, 2*3, 2*5, 3*5
- three number multiplication: 2*2*3, 2*2*5, 2*3*5
- four number multiplication: 2*2*3*5  
Then, how many unique multiplication result I can generate for following set 
{ 11, 11, 11, 11, 11, 7, 7, 7, 7, 7, 5, 5, 5, 5, 3, 3, 3, 2, 2, 2}

What is the formula?
*Edit: only prime numbers are allowed in the set

Every positive integer has a unique prime factorization. So your products will all be of the form
$$ 11^a\cdot 7^b\cdot 5^c\cdot 3^d\cdot 2^e $$
where $0\le a,b,c,d,e$ and $a,b\le 5;\; c\le 4;\; d,e\le 3$. Which gives you
$$ 6\cdot 6\cdot 5\cdot 4\cdot 4 = 2880 $$
possible numbers of this form. But that includes the case of zero factors (in which case you have $a=b=c=d=e=0$ and the product is $1$) as well as the cases of one factor ($a+b+c+d+e=1$, $5$ alternatives). Therefore your actual number is
$$ 6\cdot 6\cdot 5\cdot 4\cdot 4 - 1 - 5 = 2874 $$
This generalizes to other multisets in an obvious way. If prime number $i$ (its actual value is irrelevant) occurs $k_i$ times, and you have $n$ such factors in total, then the number of distinct non-trivial products is
$$ (k_1+1)\cdot(k_2+1)\cdots(k_n+1) - 1 - n =
\left(\prod_{i=1}^n(k_i+1)\right) - 1 - n $$
