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 ?
DETAIL शब्द के अक्षरों को कितने अलग-अलग तरीकों से इस तरह व्यवस्थित किया जा सकता है कि स्वर केवल विषम स्थानों पर ही हों?

Options

A. 32
B. 48
C. 36
D. 60

Answer

To solve the problem of arranging the letters of the word "DETAIL" such that the vowels occupy only the odd positions, we can follow these steps:

### Step 1: Identify the letters and their types
The word "DETAIL" consists of 6 letters:
- Vowels: E, A, I (3 vowels)
- Consonants: D, T, L (3 consonants)

### Step 2: Determine the positions available for vowels
In a 6-letter arrangement, the positions are numbered as follows:
1. Position 1 (odd)
2. Position 2 (even)
3. Position 3 (odd)
4. Position 4 (even)
5. Position 5 (odd)
6. Position 6 (even)

The odd positions available are 1, 3, and 5. Therefore, we have 3 odd positions for the 3 vowels.

### Step 3: Arrange the vowels in the odd positions
Since we have 3 vowels (E, A, I) and 3 odd positions (1, 3, 5), we can arrange the vowels in these positions. The number of ways to arrange 3 vowels is given by the factorial of the number of vowels:

Number of arrangements of vowels = 3! = 6

### Step 4: Arrange the consonants in the remaining positions
The remaining positions (2, 4, and 6) will be filled with the consonants (D, T, L). The number of ways to arrange these 3 consonants is also given by the factorial of the number of consonants:

Number of arrangements of consonants = 3! = 6

### Step 5: Calculate the total arrangements
To find the total number of arrangements where the vowels occupy only the odd positions, we multiply the number of arrangements of vowels by the number of arrangements of consonants:

Total arrangements = (Number of arrangements of vowels) × (Number of arrangements of consonants)
Total arrangements = 3! × 3! = 6 × 6 = 36

Thus, the total number of different ways the letters of the word "DETAIL" can be arranged such that the vowels occupy only the odd positions is **36**.

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