NDA Math Permutations and Combinations

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).
=

6!

2!

=

3!×4!

2!

=

4!

2!

(5×6−3!)
=12(30−6)
=12×24
=288