In how many different ways can the letters of the word 'DETAIL' be arranged such that the vowels must occupy only the odd positions?
Choose the correct choice from the following options: A) 12 B) 36 C) 45 D) 96 E) 83
The word 'DETAIL' has 6 letters which has 3 vowels (EAI) and 3 consonants(DTL)   The 3 vowels(EAI) must occupy only the odd positions. Let's mark the positions as (1) (2) (3) (4) (5) (6). Now, the 3 vowels should only occupy the 3 positions marked as (1),(3) and (5) in any order.   Hence, number of ways to arrange these vowels   = 3P3 =3!=3×2×1=6   Now we have 3 consonants(DTL) which can be arranged in the remaining 3 positions in any order. Hence, number of ways to arrange these consonants   = 3P3=3!=3×2×1=6   Total number of ways   = number of ways to arrange the vowels × number of ways to arrange the consonants   =6×6=36   Answer: 36
<result>B</result>
In how many different ways can the letters of the word 'DETAIL' be arranged such that the vowels must occupy only the odd positions?