How many words can be formed out of the letters of the word "ORIENTAL' so that vowals a always occupy the odd places?

Options

A. 576
B. 578
C. 676
D. 720

Answer

To solve the problem of how many words can be formed from the letters of the word "ORIENTAL" such that the vowels always occupy the odd places, we can follow these steps:

### Step 1: Identify the letters in the word "ORIENTAL"
The word "ORIENTAL" consists of 8 letters:
- Vowels: O, I, E, A (Total = 4 vowels)
- Consonants: R, N, T, L (Total = 4 consonants)

### Step 2: Determine the positions available
In an 8-letter word, the positions are numbered from 1 to 8. The odd positions are:
- 1st, 3rd, 5th, and 7th positions (Total = 4 odd positions)

### Step 3: Place the vowels in the odd positions
Since we have 4 vowels and 4 odd positions, we can place the vowels in these positions. The number of ways to arrange the 4 vowels is given by the factorial of the number of vowels:
- Number of arrangements of vowels = 4! = 4 × 3 × 2 × 1 = 24

### Step 4: Place the consonants in the remaining positions
The remaining positions (2nd, 4th, 6th, and 8th) are for the consonants. We have 4 consonants to place in these 4 positions. The number of ways to arrange the 4 consonants is also given by the factorial of the number of consonants:
- Number of arrangements of consonants = 4! = 4 × 3 × 2 × 1 = 24

### Step 5: Calculate the total arrangements
To find the total number of arrangements of the letters in the word "ORIENTAL" with the vowels in 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 = 24 × 24 = 576

### Final Answer
The total number of words that can be formed is **576**.

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