Example 3: Find the number of permutations of the letters of the word ‘REMAINS’ such that the vowels always occur in odd places.

Solution: The word ‘REMAINS’ has 7 letters.

There are 4 consonants and 3 vowels in it.

Writing in the following way makes it easier to solve these type of questions. Here are the letters laid out. For now, the numbers inside the parentheses just order them.

(1) (2) (3) (4) (5) (6) (7)

Here's the logic: the number of ways 3 vowels can occur in 4 different places = 4P3 = 24 ways.

After 3 vowels take 3 places, number of ways 4 consonants can take 4 places = 4P4 = 4! = 24 ways.

Therefore, total number of permutations possible = 24*24 = 576 ways.