5 rings on 4 fingers.

I think it's time to post my own solutions:

1. In how many ways can 5 rings be worn on the four particular fingers of the right hand?
(A) 4^5
(B) 5^4
(C) 8C3
(D) 8P3
(E) 4!*5!

OA for this question is 4^5 with OE as Economist showed in his post. But I do think that it's not correct, see my reasoning of it in my previous post.

We have 5 identical rings and 4 particular fingers. We want to count how in how many ways these 5 rings can be worn on 4 fingers. Basically it's the same as 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, which is:
n+r-1Cr-1.

In our case n=5 and r=4: (5+4-1)C(4-1)=8C3=56

Answer: C.


2. In how many ways can 5 different rings be worn on the four particular fingers of the right hand?
(A) 4^5*5C4
(B) 5^4*5C4
(C) 8C3
(D) 8P3
(E) 4!*5!

This case says that rings are not identical, as n rings on the particular finger can be worn in n! ways, we should multiply 8C3 by 5! which is 8!/3!=6720. We don't have this answer choice. But I really think this should be the answer.

Consider this:
First ring can be worn in 4 ways;
Second ring can be worn in 5 ways (as it can gon on any of four fingers - 4 ways; plus it can go below the first one - 1);
Third ring can be worn in 6 ways (the same as for the second);
Fourth ring can be worn in 7 ways;
Fifth ring can be worn in 8 ways;

Total: 4*5*6*7*8=6720.

The answer choice D should be 8P5 instead of 8P3.

3. In how many ways can 5 different rings be worn on the any four fingers of the right hand?
(A) 4^5*5C4
(B) 5^4*5C4
(C) 8C3*5C4
(D) 8P3*5C4
(E) 4!*5!*5C4

Here we should just multiply previous answer by 5C4 the number of selection of four fingers out of five: 6720*5C4=6720*5=33600.

The answer choice should be 8P5*5C4 instead of 8P3*5C4.