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  • In how many ways can the letters of the word "ENCAPSULATE" be arranged to form a word where the four vowels are in odd positions?  <div class

In how many ways can the letters of the word "ENCAPSULATE" be arranged to form a word where the four vowels are in odd positions?

 

Option: 1

78,952

 


Option: 2

60,560

 


Option: 3

28,562

 


Option: 4

40,320,


Answers (1)

To determine the number of ways the letters of the word "ENCAPSULATE" can be arranged to form a word where the four vowels (E, A, U, A) are in odd positions, we can follow these steps:

We have a total of 12 letters in the word "ENCAPSULATE".

Out of these 12 letters, 4 are vowels (E, A, U, A) and 8 are consonants (N, C, P, S, L, T).

The four vowels must be placed in the odd positions, which are positions 1,3,5, and 7 .

We can choose the positions for the four vowels in \mathrm{ C(4,4)=1} way, as they are fixed in the odd positions.

We can arrange the remaining 8 letters ( 4 consonants and 4 vowels) in the remaining even positions (2,4,6,8,9,10,11,12) in

\mathrm{ 8 !=40,320} ways

Therefore, the total number of ways the letters of the word "ENCAPSULATE" can be arranged to form a word with the four vowels in odd positions is 1 \times 40,320=40,320.

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.

Posted by

Ramraj Saini

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