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  • Determine the number of distinct words that can be formed using the letters of "ENCAPSULATE" such that four vowels occupy the odd places.  <d

Determine the number of distinct words that can be formed using the letters of "ENCAPSULATE" such that four vowels occupy the odd places.

 

Option: 1

1260


Option: 2

1440


Option: 3

3260


Option: 4

9852


Answers (1)

best_answer

To determine the number of distinct words that can be formed using the letters of "ENCAPSULATE" such that four vowels ( E, A, U, A) occupy the odd places, we can follow these steps:

There are a total of 12 letters in "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 positions, which are positions 1, 3, 5, and 7 .

The remaining eight positions (2,4,6,8,9,10,11,12) can be filled with the remaining eight letters (three consonants and two vowels) in different arrangements.

We can choose three consonants from the seven available consonants in \mathrm{C(7,3)=35} ways.

We can choose two vowels from the three available vowels in \mathrm{ C(3,2)=3} ways

The three chosen consonants can be arranged in the remaining three even positions in \mathrm{ 3 !=6} ways.

The two chosen vowels can be arranged in the remaining two even positions in 2 !=2 ways.

Therefore, the total number of distinct words that can be formed is 35 \times 3 \times 6 \times 2=1,260.

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

Posted by

HARSH KANKARIA

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