Question

The number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated, is:
$
  {\text{A}}{\text{. 4!4!}} \\
  {\text{B}}{\text{. }}\dfrac{{8!}}{{4!}} \\
  {\text{C}}{\text{. 288}} \\
  {\text{D}}{\text{. None of these}} \\
$

Answer 1

Hint: To find the number of ways of picking 8 flowers such that 4 are never separated, we use the formula of arranging objects in a circular order. As four flowers are not separated, it looks like we are arranging 5 objects in total. We calculate the number of ways of arranging 5 objects and the number of ways the flowers are arranged internally.

Complete step-by-step answer:
Given Data,
8 different flowers can be strung to form a garland so that 4 particular flowers are never separated

It is given that four flowers are not separated, so these four flowers together look like a single entity as we cannot separate them.
So now the total number of objects to be arranged is 5, 4 flowers and 1 entity having four flowers.

We know the formula of arranging objects in a circular manner is (n – 1)!
The number of ways of arranging 5 objects in a circular order is (5 – 1)! = 4!

Now we can rearrange the flowers inside of the inseparable entity, so the number of ways of arranging four flowers inside the entity is, n! = 4!

Therefore, the total number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated, is 4!4!
Option A is the correct answer.


Note – In order to solve questions of this type the key is to understand we would essentially have to arrange 5 objects in a garland only, as four flowers are separable. We have to know the formula of arranging n objects in a circular order. Also, we have to know the formula of arranging n objects in n spaces and other basic formulae of permutations and combinations.
The factorial of a number n represented by n! is equal to n! = n (n - 1) (n – 2) …….. (n – (n – 1)).