Find number of integral solutions of a*b*c*d = 600 The number of ordered solutions comes out to be 800. I need to find the number of distinct solutions but I'm stuck at calculating the possible combinations.
Any ideas on how to proceed further?
 A: The prime factors of $600$, expressed as a multiset, are $\{2,2,2,3,5,5\}$. To get the original answer of $800$ ordered tuples, we partition each distinct prime across $(a,b,c,d)$ with stars & bars to get: $${6 \choose 3}{4 \choose 3}{5\choose 3} = 20\cdot 4\cdot 10 = 800$$
Clearly we can never have all four factors the same. We can have three factors the same only in $2$ distinct cases, $(1,1,1,600)$ and $(2,2,2,75)$, which correspond to $8$ choices in the $800$. 
Two factors the same, $(a,a,X,Y)$, can have $a \in \{1,2,5,10\}$. For $a=1$ in given position, there are ${4 \choose 1}{2 \choose 1}{3\choose 1}=24$ options, and similarly $a=(2,5,10)$ have $(12,8,4)$ options. For $a=(1,2)$ we also need to reduce the values by $2$ to remove the overlap with the three-equal-factor case. The total of $44$ options correspond to $44\cdot {4 \choose 2} = 264$ cases in the $800$ but only $22$ in the distinct cases.
The remaining cases, $800-264-8 = 528$, have four distinct values and are a multiple of $4!$ of the distinct cases, giving $528/24 = 22$ more distinct cases, for a total of:
$$ 2+22+22 = 46 \text{ distinct cases}$$
