In how many ways can 5 different rings be worn in four

I think you are overthinking the concept by introducing the "identical basket" situation. For the first case if, for example, there is no difference between the following scenarios: {5-0-0-0} and {0-5-0-0}, then you can manually write down all possible cases: {5-0-0-0}, {4-1-0-0}, {3-1-1-0}, {3-2-0-0}, {2-2-1-0}, {2-1-1-1}. So the answer for the first question as you stated would be 6.

Don't overcomplicate it: the GMAT combination/probability questions are fairly straightforward and no need to waste time on the problems you will never see on the test.

If the questions were:

1. In how many ways can 5 identical fruits be distributed in 4 different baskets?

Consider five stars and three bars: *****|||, where stars represent fruits and three bars (one less than the baskets) will help us to divide them among the baskets. How many permutations (arrangements) of these 5+3=8 symbols are there? Permutation of 8 symbols out of which 5 * and 3 | are identical is 8!/(5!3!). Now, each arrangement will mean different scenario for fruit distribution. For example: **|*|*|* will mean that the first basket gets 2 fruits, the second, third and fourth 1 fruit. Or: ***|*|| will mean that the first basket gets 3 fruits, the second 1, and third and fourth none, and so on.

Answer: 8!/(5!3!).

2. In how many ways can 5 different fruits be distributed in different 4 baskets?

This one is easier each of the 5 different fruits has 4 choices, so total # of distribution is 4*4*4*4*4=4^5.

Answer: 4^5.

Check similar questions:
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Theory on permutation and combinations:
permutation-86687.html

Direct formula if needed:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is \(n+r-1C_{r-1}\).

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is \(n-1C_{r-1}\).

Hope it helps.