Consider the word NATION There are 2 N’s and in total 3 vowels. Let us consider all the three vowels (AIO) as one pack. Then there will be in total 4 articles (two N’s, one T, and one pack of vowels). The number of ways in which these 4 articles can be arranged will be =
4!
2!
Now the vowels can be internally arranged in 3! ways. Hence the number of words formed in which the vowels are always together =
3!×4!
2!
Therefore the total number of words formed from the letters of the word NATION, where the vowels are not together will be equal to (Total number of words formed) - (Total number of the words where the vowels are together). =