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In how many different ways can the letters of the word “DETAIL” be arranged in such a way that the vowels occupy only the odd positions?

In how many different ways can the letters of the word “DETAIL” be arranged in such a way that the vowels occupy only the odd positions? Correct Answer 36

There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants

Let the positions be marked as (1)(2)(3)(4)(5)(6)

The 3 vowels can be placed at any of the 3 places marked (1)(3)(5)

The number of ways of arranging the vowels = 3P3 = 3! = 6

Also, the 3 consonants can be arranged in the remaining 3 places

The number of ways of these arrangements = 3P3 = 3! = 6

Total number of ways = 6 × 6 = 36

∴ Total number of ways = 36

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