Question

The number of ways in which $ 8 $ red roses and $ 5 $ white roses of different sizes can be made out to form a garland so that no two white roses come together is
A. $ \dfrac{{8!}}{{2!}}.{}^8{P_5} $
B. $ \dfrac{{7!}}{{2!}}.{}^8{P_5} $
C. $ \dfrac{{7!}}{{2!}}.{}^9{P_5} $
D. $ 7!.{}^4{P_3} $

Answer 1

Hint: Here we are asked to find the number of ways in which the roses can be placed in such a way that no white roses are placed together. There will be eight spaces between the eight red roses and place five white roses in it. Here we will use the concepts for the circular permutations to find the total number of ways.

Complete step-by-step answer:
Permutations can be defined as the number of ways in which the objects of the set can be selected keeping the order of the objects important.
Given that eight red roses and five white roses are different sizes and are arranged in the garland so it is in the circular form.
The total number of ways in which “n” objects can be arranged in the circular form is given by –
 $ \dfrac{{(n - 1)!}}{2} $
Place the value for the eight red roses –
Total number of ways $ = \dfrac{{(8 - 1)!}}{2} $
Simplify the above expression –
Total number of ways $ = \dfrac{{7!}}{2} $ ways ….(A)
Also, given that no two white roses should be placed together in these eight spaces so by using the formula for permutations,
 $ {}^n{P_r} = \dfrac{{n!}}{{(n - r)!}} $
Place the values in the above expression –
 $ {}^8{P_5} = \dfrac{{8!}}{{(8 - 5)!}} $ ………. (B)
From the equations (A) and (B)
The total number of ways to place all the roses in the circle are $ \dfrac{{7!}}{{2!}}.{}^8{P_5} $
Hence, from the given multiple choices – the option B is the correct answer.
So, the correct answer is “Option B”.

Note: Permutations are expressed as the different ways of arranging elements in the certain favorable pattern. Here we were given roses of different sizes and therefore have used permutation. Do not get confused between the permutations and combinations concepts and apply it accordingly.