Question

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

• A. $354$
• B. $348$
• C. $288$
• D. None of the above

Answer 1

Answer: (D)

None of the above

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

$\therefore$ 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 $=360-144=216$