Wow so many rings and fingers.
I agree the answer to the original question is 4^6 if the rings are different from each other. It may be more complicated if the order of the rings on one finger is important. Would it be 4^6*P(6,6)?
If they are not different from each other, then we need to separate the 6 rings into four sections with three divisors. Total number of positions is 6+3=9 for 6 rings and 3 divisors. We need to pick three positions to place the three divisors. So the outcome would be C(9,3).
Now the different variations:
1) How many ways 6 rings can be worn in 4 fingers when a finger can have a maximum of 2 rings?
Assuming they are not different from each other. There are 6 rings and 4 fingers so that means there could only be one finger without a ring if each finger can have a maximum of 2 rings. The outcome would be C(4,1) to decide which finger would be the bare finger, then the other three fingers would have 2 rings each.
Another type of outcome would be no bare fingers. In other words each finger would have at least 1 ring, then we need to pick two fingers to each have 1 more ring. The outcome would be C(4,2).
So my answer to this question is C(4,1)+C(4,2)
If they are different from each other, then order would become important too.
One bare finger: C(4,1)*P(6,6)
No bare fingers: C(4,2)*P(6,6)
2) How many ways 6 rings can be worn in 4 fingers when a finger can have a maximum of 3 rings and each finger must have at least 1 ring?
Assuming they are not different. Each finger must have 1 ring so we only need to arrange the 2 extra ring. The total outcome would be C(5,3)=10. You could also do it this way: the two rings are in one finger: C(4,1). The two rings are in different fingers: C(4,2). Total outcomes=4+6=10.
If they are different, then we would need to take out the case where there is one or more bare fingers from the total outcome 4^6. This will also take care the maximum 3 condition.
3) How many ways 6 rings can be worn in 4 fingers when at least 3 fingers should have rings and any finger can't have more than 2 rings?
Isn't it the same with question number 1)?