In how many ways can $6$ different rings be placed on $4$ fingers?
Atempt:
Using the ball-dash method to decide how many rings there will be on each finger, we get $x_1+x_2+x_3+x_4=6$, which yields $84$ solutions. We have $\binom{10}{6}$ ways to select $6$ from $10$. So we have $84\cdot 210$. But the answer is $60480$. I would like to know what my mistake
Best Answer
If the rings are indistinguishable, then a stars and bars argument indeed gives $\binom{9}{3} = 84$ ways to assign rings to fingers. However, it isn't quite right to then multiply this by $\binom{10}{6}$, which does not have an combinatorial significance in the problem.
To obtain the desired answer, we need to count the number of ways the rings can be put onto the fingers, assuming the rings are all distinct and that the order in which they are placed on the fingers matter. We should consider all $6! = 720$ ways to order the rings and then all $\binom{9}{3} = 84$ ways to distribute the rings among the fingers such that they appear in a given order. Thus, there are a total of $720 \cdot 84 = 60480$ ways to put the rings on the fingers.