How many words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together?

Question

Filo · Accepted Answer

Number of words formed so that all the three vowels are never together = Total number of words formed using all the letters in the word 'NATION'

-

Number of words where all the vowels come together

Number of letters repeating in the word

= 2

(N's)

Vowels in the word are

A, I, O

Total number of words that can be formed using the letters in the word 'NATION'

= \frac{6 !}{2 !} = 360

Consider

A, I, O

as one group

Then the no. of words formed by this group and remaining letters is

4!

The three vowels can be arranged among themselves in

3!

ways

∴

Number of words where all the vowels come together

= (4!) (3!) = 144

Thus, the number of words formed so that all the three vowels are never together

= 360 - 144 = 216

Question Text	How many words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together?
Answer Type	Text solution:1
Upvotes	150

Text solutionVerified