Question: The number of ways in which 8 different flowers can be seated to form a garland so that particular four flowers are never separated is
- 4! * 4!
- 288
- 8! / 4!
- 5!* 4!
- 8! * 4!
Correct Answer: B
Solution and Explanation:
Approach Solution 1:
It is asked to find out the number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated.
According to the question, there are 8 flowers, but it is given that four particular flowers are never separated. So four particular flowers will always remain adjacent to each other.
To solve this problem we should know the following cases of circular permutations:
- If clockwise and anti-clockwise orders are different, the total number of circular permutations is: (n-1)!.
- If anti-clockwise and clockwise orders are taken as the same, then the total number of circular permutations is given by (n-1)!/2!
It is important to know which formula is applicable in this case.
When we turn a garland it forms a different arrangement. But it still is the same garland.
For example, consider a set of numbers like ‘1234’. When turned around it looks like ‘4321’ but it still is the same permutation of numbers.
In a similar way, the garland when turned might have a different permutation but it still is the same garland.
Therefore we’ll use the 2nd formula.
For n distinct objects in which clockwise and anticlockwise orders are taken as the same, the total number of circular permutations is: (n-1)!/2
Here in this case, given that four flowers must be together in all cases.
So we’ll club together all four particular flowers and consider this as one object.
Now we have 5 different objects - four flowers and four particular flowers clubbed as one object. Now we have to arrange these five objects in five circular seats.
In that case, the number of permutations is (5-1)!
And in every permutation, the clubbed flowers can be arranged in 4! ways.
So there are a total of (5-1)! * 4! Permutations.
But we have seen that the garland is the same in anticlockwise and clockwise order.
Therefore for every permutation, we have included its rotated permutation.
So the total number of circular permutations of garlands is : (5-1)!*4! /2
(5-1)! * 4! /2 = (4*3*2*1)*(4*3*2*1)/2
= 24 * 12 = 288
Thus the total number of ways in which 8 different flowers can be seated to form a garland so that four particular flowers are never separated is 288.
Approach Solution 2:
It is asked to find out the number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated.
This question is related to Permutations and combinations
Let's take the 4 flowers which are never separated as a set.
Let’s consider 4 flowers in a set as 4 elements.
Now, the flowers arranged in a number of ways = 4!
Furthermore, 4 flowers in a set arranged in a number of ways = 4!
Therefore, the total number of ways = 4! × 4!
We can arrange the flowers in both directions i.e clock and anti-clockwise.
Hence, actual number of ways = ( 4! × 4! ) / 2
= ( 4 × 3 × 2 × 4 × 3 × 2 ) / 2
= 4 × 3 × 2 × 4 × 3
= 288
Thus the total number of ways in which 8 different flowers can be seated to form a garland so that four particular flowers are never separated is 288.
“The number of ways in which 8 different flowers can be seated to form”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The GMAT Problem Solving questions enable the candidates to consider every piece of data in order to solve the numerical problems. GMAT Quant practice papers enable the candidates to analyse several types of questions that will enable them to improve their mathematical knowledge.
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