Number of ways in which 6 rings can be worn on the 4 fingers of one hand The way I solved this is - 
The 1st finger can have any of the 6 rings, $\therefore 6$ ways
The 2nd finger can have any of the 5 remaining rings, $\therefore 5$ ways
The 3rd finger can have any of the 4 remaining rings, $\therefore 4$ ways
The 4th finger can have any of the 3 remaining rings, $\therefore 3$ ways
$\therefore$ total number of ways = 5*4*3*2 = 120 ways.
But every website I check for the solution, they are different and I'm very confused. Is my approach correct?
 A: You can order the $6$ rings in $6! = 720$ ways. Now you have to split the string of $6$ rings into $4$ groups, one for each finger. One way to see how to do this is stars and bars, you have $6$ stars ($*$, the rings) and $4 - 1 = 3$ bars ($\mid$, separations between groups), this is to select $4 - 1$ positions for the $\mid$ among $6 + 4 - 1$ positions in all. Pulling all together:
$\begin{align}
6! \cdot \binom{6 + 4 - 1}{4 - 1} = 720 \cdot 84 = 60480
\end{align}$
A: See, fixing the 4 fingers and permutation over the rings doesn't work in these cases.
So, just fix the 6 rings
r1, r2, r3, r4, r5, r6
Now, first ring can be worn in four ways (in either of the 4 fingers) , similarly 2nd ring also in 4 ways..
So total ways are 4×4×4×4×4×4 = 4096
I hope this helps..
A: There are simply 4 ways for each of the 6 rings to be placed in the fingers. So, 1 answer is 4^6 = 4096. But this can only be used if the rings are not identical.
If the rings are identical then, stars and bar method is the best. We can place the 6 rings with the 3 bars in a line. Now there are 9C3 ways of arranging the rings. Hence, there are 84 ways of placing the rings if the rings are identify.
Another case is when we are just to use the 4 out of 6 rings each in a finger and the rings are not identical. Here we can use the formula of 654*3 that is 360.
I hope your confusion is cleared.
A: See you can choose the fingers in ${5\choose 4}$ ways . and then normal calculations which you have done $=5.6.5.4.3=1800$ hope its clear.
