Question

The number of different ways that three distinct rings can be worn to 4 fingers with at most one ring in each of the fingers is:
A. ${}^4{P_3}$
B. ${}^5{P_4}$
C. ${}^5{P_3}$
D. ${}^4{P_2}$

Answer 1

Hint: We will proceed towards the final solution assuming the number of rings which can be worn in the first finger to be the maximum and then consecutively about the second ring and so on and after that we will use the permutations method.

Complete step-by-step answer:

Here, we are given that three rings are to be worn in different ways in 4 fingers and the condition is given that at most one ring can be worn in each finger.
Considering three rings, a person can wear the first ring in any of the four fingers which means there are 4 ways in which the first ring can be worn.
Now for the second ring, when the first ring is already been worn in any one finger that means there are only three fingers left in which we can wear the second ring. Hence, there are only 3 ways in which the second ring can be worn.
For the third and the last ring, we are left with only two fingers as any two are already occupied with the first two rings. So, there are 2 ways in which we can wear the third ring.
Hence, the total number of ways in which we can wear the rings are: $4 \times 3 \times 2 = 24$
This implies that rings can be worn in 24 ways.
But 24 ways is not an option. Hence, we will check which option is the same as that of 24.
Option(A): ${}^4{P_3}$.
We know that the formula of permutations of n things chosen r at a time is given by: $P\left( {n,r} \right) = {}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
 $ \Rightarrow {}^4{P_3} = \dfrac{{4!}}{{(4 - 3)!}} = \dfrac{{4!}}{{1!}} = \dfrac{{4 \times 3 \times 2 \times 1}}{1} = 24$
Therefore, we can say that option(A) is correct.

Note: In such questions, we first sketch the possible ways, and then we check if we need to multiply them or add them. We multiply when we are not done with a given situation. We add only when we are given an extra condition to consider. For e.g., if I add the condition in the above question that also finds the total number of ways in which the rings can be worn with and without repetition, then we will add the ways in which the rings can be worn with repetition in our answer.