Question

A bangle is studded with $ 18 $ pearls of different colors. What is the total number of ways in which the pearls studded in the bangle can be arranged in such a way that there is always one pearl between a red and a blue pearl?
(A) $ 2 \times 19! $
(B) $ 2 \times 18! $
(C) $ 18 \times 19! $
(D) $ 2 \times 16! $

Answer 1

Hint: In the given question, we are given a bangle with eighteen pearls of different colors studded in it. We are to calculate the number of ways such that there is always a pearl in between the pearls of colors blue and red. The question revolves around the concepts of permutations and combinations. One must know about the fundamental principle of counting in order to solve such types of questions. According to the fundamental principle of counting, if there are p number of ways to do one thing and q number of ways to do another thing, then there are pq number of ways of doing both the things together.

Complete step by step solution:
So, we are given the condition that there is always a pearl in between the blue and red pearls.
Firstly, the number of ways in which red and blue pearls can be arranged is $ 2! = 2 $ .
Now, there are a total of $ 16 $ pearls of different colors.
Out of these sixteen pearls, we have to pick up one to place between the red and blue pearl.
So, the number of ways of selecting one pearl out of the remaining $ 16 $ pearls is
$ ^{16}{C_1} = 16 $ .
Now, we have to arrange the remaining $ 15 $ pearls in the bangle.
So, there is $ 15! $ ways of arranging $ 15 $ different pearls in a bangle.
Now, using the fundamental principle of counting, we get,
The total number of ways in which the pearls can be studded in the bangle are:
$ 2 \times 16 \times 15! $ .
Simplifying the expression to match the options, we get,
 $ \Rightarrow 2 \times 16! $
Therefore, the total number of ways in which the pearls studded in the bangle can be arranged in such a way that there is always one pearl between a red and a blue pearl is $2 \times 16! $. So, the correct option is Option (D).

Note: The question revolves around the concepts of Permutations and Combinations. One should know about the principle rule of counting or the multiplication rule. Care should be taken while handling the calculations. Calculations should be verified once so as to be sure of the answer.