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  • How many different words can be formed with the letters of the word "ENCAPSULATE" such that: (i) Four vowels occupy the odd places? (ii) The word begins with the letter N? (iii) The word

How many different words can be formed with the letters of the word "ENCAPSULATE" such that: (i) Four vowels occupy the odd places? (ii) The word begins with the letter N? (iii) The word begins with the letter E and ends with the letter T?

 

Option: 1

(i) 86,400 (ii) 3,628,800 (iii) 362,880


Option: 2

(i) 56,406 (ii) 5,627,800 (iii) 872,670


Option: 3

(i) 96,350 (ii) 8,528,800 (iii) 962,380


Option: 4

(i) 26,405 (ii) 4,668,800 (iii) 762,880


Answers (1)

best_answer

To find the number of different words that can be formed with the letters of the word "ENCAPSULATE" given the conditions, let's analyze each condition separately:

(a) Four vowels occupy the odd places:

The vowels in the word "ENCAPSULATE" are E, A, U, and E. There are 4 vowels in total. Since there are 5 odd places (positions 1, 3, 5, 7, 9), we need to select 4 out of the 5 odd places for the vowels to occupy.

The number of ways to select 4 out of 5 odd places is given by the combination formula: C(5, 4) = 5.

For each selected arrangement of the odd places, we can arrange the vowels in those positions in 4! ways. Similarly, we can arrange the consonants in the remaining even positions in 6! ways.

Therefore, the total number of different words that can be formed is
5 \times 4 ! \times 6 !=5 \times 24 \times 720=86,400.

(b) The word begins with the letter N:

To form a word that begins with the letter N, we fix the letter N at the beginning and arrange the remaining 10 letters (E, C, A, P, S, U, L, A, T, E).

The remaining 10 letters can be arranged in 10 ! ways.
Therefore, the number of different words that can be formed is 10 !.
 

(c) The word begins with the letter E and ends with the letter T:

To form a word that begins with the letter E and ends with the letter T, we fix the letters E and T at the beginning and end, and arrange the remaining 9 letters (N, C, A, P, S, U, L, A, E).

The remaining 9 letters can be arranged in 9 ! ways.
Therefore, the number of different words that can be formed is 9 !.

So, the answers to the given conditions are:

(a) 86,400 different words can be formed with four vowels occupying the odd places.

(b) 10 ! = 3,628,800  different words can be formed with the word beginning with the letter N.
(c) 9 ! = 3,628,80 different words can be formed with the word beginning with the letter E and ending with the letter T.

 

 

 

Posted by

Deependra Verma

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