Question

The number of ways in which 6 identical rings can be worn on 4 fingers of one hand is:
A.${}^9{C_3}$
B.${}^9{C_4}$
C.${6^4}$
D.${4^6}$

Answer 1

Here we need to find the number of ways in which the given number of identical rings can be worn on 4 fingers. As these rings are identical, repetition is allowed here. So we will first find the number of ways in which the first ring can be worn on four fingers, then we will find the number of ways in which the second ring can be worn on four fingers. Similarly, we will find it for the rest of the rings. From there, we will get our required answer.

Complete step-by-step answer:
We have to find the number of ways in which 6 identical rings can be worn on 4 fingers.
As the rings are identical, repetition is allowed here.
We can wear 1 ring in four fingers in four different ways.
Number of ways to wear first ring in four fingers $= 4$
Number of ways to wear second ring in four fingers $= 4$
Number of ways to wear third ring in four fingers $= 4$
Number of ways to wear fourth ring in four fingers $= 4$
Number of ways to wear fifth ring in four fingers $= 4$
Number of ways to wear sixth ring in four fingers $= 4$
Total number of ways in which 6 identical rings can be worn on 4 fingers will equal to the product of all these.
Therefore,
Total number of ways to wear six rings in four fingers $= 4 \times 4 \times 4 \times 4 \times 4 \times 4$
We know that when the exponents with the same base are multiplied, their powers get added.
Therefore, using this property here, we get
Total number of ways to wear six rings in four fingers $= {4^{1 + 1 + 1 + 1 + 1 + 1}} = {4^6}$
Hence, the correct option is option D.

Note: We have used the properties of exponentials here. When we multiply exponents with the same base are multiplied, their powers get added. If we divide exponents with the same base, their powers get subtracted. Here we might make a mistake by adding the number of ways to find to wear six rings in four fingers instead of multiplying the, This will give us the wrong answer. So we need to be careful while finding the total number of ways.