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  • How many different words can be formed with the letters of the word "ENCAPSULATE" such that four vowels occupy the odd places?  <div class='q

How many different words can be formed with the letters of the word "ENCAPSULATE" such that four vowels occupy the odd places?

 

Option: 1

4,032,000

 


Option: 2

9,856,002

 


Option: 3

4,125,700

 


Option: 4

7,950,620


Answers (1)

best_answer

To determine the number of different words that can be formed with the letters of the word "ENCAPSULATE" such that four vowels occupy the odd places, we can consider the following:

There are a total of 12 letters in the word "ENCAPSULATE", including 5 vowels ( E, A, U, A, E) and 7 consonants (N, C, P, S, L, T).

The four vowels must occupy the odd places, which means they will be placed in positions 1,3,5, and 7.

The remaining eight letters (four consonants and four vowels) can be arranged in the even places, which means they will be placed in positions 2,4,6,8,9,10,11, and 12 .

We can choose four vowels from the five available vowels in \mathrm{C(5,4)=5} ways

We can arrange the four chosen vowels in the odd places in \mathrm{4 !=24} ways.

We can arrange the remaining eight letters (four consonants and four vowels) in the even places in 8 ! \mathrm{=40,320} ways.

Therefore, the total number of different words that can be formed is \mathrm{5 \times 24 \times 40,320=4,032,000.}

It is important to note that repetition is not allowed in this case, as we are using the letters of the word "ENCAPSULATE" only once.

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seema garhwal

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