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  • In how many different ways can the letters of the word 'STRAWBERRY' be arranged so that the vowels never come together?Option: 1 <p di

In how many different ways can the letters of the word 'STRAWBERRY' be arranged so that the vowels never come together?

Option: 1

4156780 


Option: 2

2903040


Option: 3

3104430

 


Option: 4

5904030


Answers (1)

best_answer

To find the number of different ways the letters of the word 'STRAWBERRY' can be arranged such that the vowels never come together, we can use the principle of inclusion-exclusion.

First, let's consider all the possible arrangements of the 10 letters without any restrictions. The word 'STRAWBERRY' has 10 letters, so there are 10! (10 factorial) ways to arrange them.

Next, we need to subtract the arrangements where the vowels ('A', and 'E') come together.

Let's treat the two vowels ('A', and 'E') as a single entity. This reduces the problem to arranging 9 entities (the combined vowel entity and the other 8 letters) around a line.

The number of ways to arrange 9 entities around a line is 9!

Within the combined vowels entity, the vowels 'A' and 'E' can be arranged among themselves in 2! ways.

Therefore, the number of arrangements where the vowels ('A', 'E') come together is given by,

9! \times 2!

Now, let's subtract the number of arrangements where the vowels come together from the total number of arrangements:

\text{The number of ways of arranging seven letters such that no two vowels come together} = \text{Number of ways of arranging seven letters} - \text{Number of ways of arranging the letters with the two vowels being together}\\ \begin{aligned} & 10 !-9 ! \times 2 !=3628800-725760 \\ & 10 !-9 ! \times 2 !=2,903,040 \end{aligned}

Hence, there are 2,903,040 different ways to arrange the letters of the word 'STRAWBERRY' such that the vowels never come together.

 

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Riya

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