In how many ways can the letters of the word ‘EQUATION’ be arranged so that all the vowels come together?a)9C4 . 9C5b)4!.5!c)9!/5!d)9!-4!5!Correct answer is option 'B'. Can you explain this answer?

Question

In how many ways can the letters of the word &#8216;EQUATION&#8217; be arranged so that all the vowels come together?a)9C4 . 9C5b)4!.5!c)9!/5!d)9!-4!5!Correct answer is option 'B'. Can you explain this answer?

Answer

There are five vowels.. so they can be arranged in 5! ways.
other three can be arranged in 3! ways.
now vowels have to be together so they can be at start, end or in between the other three words.. this can be done in 4 ways.
hence it can be done 5!.3!.4 =5!.4!

Answer

The answer is 5! (if vowels are taken as one unit) x 3! (the consonants)

Answer

Introduction:
The given question asks us to find the number of ways the letters of the word 'EQUATION' can be arranged such that all the vowels come together.

Approach:
To solve this problem, we will treat the vowels (E, U, A, I, O) as a single entity and arrange the remaining consonants (Q, T, N) along with this entity. Then, we will find the number of ways to arrange this combined entity and multiply it by the number of ways to arrange the remaining consonants.

Step 1: Counting the number of ways to arrange the vowels:
Since we are treating the vowels as a single entity, there are 5 vowels. We can arrange these 5 vowels among themselves in 5! (5 factorial) ways. This is because there are 5 options for the first position, 4 options for the second position, 3 options for the third position, and so on. Therefore, the number of ways to arrange the vowels is 5!.

Step 2: Counting the number of ways to arrange the consonants:
The word 'EQUATION' has 3 consonants (Q, T, N). We can arrange these 3 consonants among themselves in 3! ways. Similarly, there are 3 options for the first position, 2 options for the second position, and 1 option for the third position. Therefore, the number of ways to arrange the consonants is 3!.

Step 3: Combining the arrangements:
Now, we need to arrange the combined entity (vowels) and the remaining consonants. We have already calculated the number of ways to arrange the vowels (5!) and the consonants (3!). To find the total number of ways to arrange the word 'EQUATION' such that all vowels come together, we multiply the number of ways to arrange the vowels by the number of ways to arrange the consonants.

Total number of ways = 5! * 3!

Final Answer:
Simplifying the expression, we get:
Total number of ways = 5! * 3!
Total number of ways = 5! * 3 * 2 * 1
Total number of ways = 120 * 6
Total number of ways = 720

Therefore, the correct answer is option 'B' (4!.5!).

Similar Quant Doubts

In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together?

In how many different ways can the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together?

In how many different ways can the letters of the word ‘MATHEMATICS’ be arranged so that the vowels always come together?

Top Courses for Quant

Related Content

Quant Courses

Related Exams

Related Question Answers

Related Searches

Company