I'm not sure if I worded the question properly, but I'm trying to understand how to solve certain problems. I'll begin with things I think I understand.
Consider for example the problem of grouping $n$ balls into bags. If the balls are all identical, and we want to know how many ways we can separate them into an unordered set of bags (any number of bags), it is equivalent to integer partitioning of $n$. Here, the fact that the balls are identical, could be interpreted as: groupings are equivalent modulo any permutation of the balls
Conversely, if the $n$ balls are all distinct then the answer can be expressed using bell numbers, $B_n$, for the number of possible partitions of n distinct elements. Balls are distinct, means groupings are all distinct modulo any permutation.
Where I'm going with this, is how do you think of problems like this: If you have 1 green ball and 3 red balls {G, R, R, R}, how many distinct groupings of these balls can be made? All red balls are identical, all green balls are identical, but a green and red ball are distinct. In a sense, only some permutations are allowed to make two groupings equivalent.
The answer to the above problem is 7, but what is the combinatorial way of solving any generalization of the problem?
Thank you in advance, Alex.
P.S. I arrived at this question trying to calculate the number of factorizations of an integer given its prime factors. The above problem I think is equivalent to counting the number of factorizations of $54 = 2 \cdot 3 \cdot 3 \cdot 3$. Putting balls into a bag is the same as multiplying the corresponding primes. Thus { {G, R} {R} {R} } corresponds to $54 = 6 \cdot 3 \cdot 3$