In how many ways $5$ different rings can be worn on $4$ fingers?

Question

In how many ways $5$ different rings can be worn on $4$ fingers ?

Although these is similar question here In how many ways $3$ different rings can be worn in $4$ fingers with at most one in each finger?

But I want to answer it in different way which is like take $4$ fingers like $a,b,c,d$. Now $a$ can be filled in $5$ ways and so are others. So total ways are $5^4$ but answer is $4^5$. What is wrong in my reasoning? Which cases have I left out?

Answer 1

The answer is $4^5$ because you have $4$ fingers for each of the ring. Or, in other words each ring have $4$ choices. $$4\times4\times4\times4\times4=4^5$$

Answer 2

Suppose you had only one ring. It then would be $4^1=4$ different arrangements/placements. Now if we add one more ring, it will be $4\cdot4$ because we combine the four possible arrangements with yet another four arrangements which makes it $16$. This means you can have two rings on one finger. By the same token we would have $4^5=1024$ arrangements for five rings. And the person can have five rings on one finger or no ring at all on the finger.

If you had $5^4$ arrangements, it would mean $5^1=5$ different arrangements for one ring on four fingers. And that doesn’t make sense.

And finger $a$ can have six possible arrangements: either of five rings or no ring at all.

Hope my explanation was not confusing.

Answer 3

I'm assuming that the four fingers are given, e.g., the fingers of one hand without the thumb. An arrangement of five distinguishable rings on the four labeled fingers amounts to a linear arrangement of $1$, $2$, $3$, $4$, $5$, and three indistinguishable zeros as separators. There are ${8!\over3!}=6720$ such arrangements. Note that a green ring and a blue ring on the index finger can be worn in $2$ ways.

Answer 4

There are 5 different rings . But you want 4 rings . Hence , there are 5C4 ways to select the rings . Now , assuming atmost and atleast 1 ring is worn in one finger , total number of ways of selecting the fingers is 5C4 . Now , once they are selected , notice that there are 4P4 i.e.(4!) Ways to arrange your selections within the fingers . Hence total arrangements shall be 5C4 * 5C4 * 4P4 = 600

Answer 5

The correct option is B 6720 Since rings are distinct, hence they can be named as R 1 , R 2 , R 3 , R 4

a n d

R 5 .

The ring R 1 can be placed on any of the four fingers in 4 ways. The ring R 2 can be placed on any of the four fingers in 5 ways since the finger in which R 1 is placed now has 2 choices, one above the R 1 and one below the ring R 1 . Similarly R 3 , R 4 and R 5 can be arrange in 6, 7 and 8 ways respectively. Hence, the required number of ways

4 × 5 × 6 × 7 × 8

6720