Question

The number of ways in which 20 different pearls of two colours can be set alternately on a necklace, there being 10 pearls of each colour, is
A. $ 9!\times 10! $
B. $ 5{{(9!)}^{2}} $
C. $ {{(9!)}^{2}} $
D. none of these

Answer 1

Hint: First, find the number of ways to arrange 10 pearls of the same colour on the necklace. This will be a circular arrangement. Then find the number of ways to arrange the remaining 10 pearls between them.

Complete step-by-step answer:
It is given that a necklace is to be made of 20 different pearls. The pearls are of two types of colours (say red and blue), This means that 10 pearls are of red colour and the other 10 are of red colour. Now, we have to set the pearls of the same type alternately and it is asked to calculate the number of ways this can be done.
Since the 20 pearls must be arranged alternately (with respect to colour), we can first arrange 10 pearls of the same colour first and then arrange the remaining 10 between them.
The number of ways to arrange ‘n’ different objects in a circular manner is equal to $ (n-1)! $ .
Here, $ n=10 $ .
Therefore, the number of ways in which 10 pearls of the same colour can be arranged on the necklace is $ (10-1)!=9! $ .
Now, we shall find the number of ways for arranging the remaining 10 pearls in between the first 10 pearls.
Once the first 10 pearls are set on the necklace, there are 10 slots made between the pearls. From the concept of permutation, we know that the number of ways in which ‘n’ different objects can be arranged in ‘n’ slots is equal to $ n! $
Therefore, the number of ways for arranging the remaining 10 pearls in between the first 10 pearls is $ 10! $
However, there will be some pairs of arrangements where they will be in clockwise and anticlockwise rotation. And this type of arrangement will appear the same in the case of a necklace.
Therefore, the total number of ways to arrange the pearls according to the given condition is $ \Rightarrow \dfrac{9!\times 10!}{2}=\dfrac{9!\times 10\times 9!}{2}=5(9!) $ .
Hence, the correct option is B.
So, the correct answer is “Option B”.

Note: If you got confused and did not understand the above solution, then you can select any two colours for the pearls, say red and blue. Then to understand the problem, take a less number of pearls, say 6 (3 red and 3 blue).
You can experiment by arranging the 3 red pearls first and then the 3 blue pearls between them. In this you will observe that they are arrangements that are clockwise and anticlockwise rotated pairs. But they cannot be considered as two different arrangements because it is the same in a necklace.