How many words can be formed of the letters in the words RAINBOW , so that the vowels may occupy only even positions ?

Options

A. 567
B. 144
C. 576
D. 625

Answer

To solve the problem of how many words can be formed from the letters in the word "RAINBOW" such that the vowels occupy only even positions, we can follow these steps:

### Step 1: Identify the letters in "RAINBOW"
The word "RAINBOW" consists of 7 letters: R, A, I, N, B, O, W. Among these:
- **Vowels**: A, I, O (3 vowels)
- **Consonants**: R, N, B, W (4 consonants)

### Step 2: Determine the positions available
In a 7-letter word, the positions are numbered as follows:
1. 1st (odd)
2. 2nd (even)
3. 3rd (odd)
4. 4th (even)
5. 5th (odd)
6. 6th (even)
7. 7th (odd)

The even positions available are 2, 4, and 6.

### Step 3: Place the vowels in the even positions
We need to place the 3 vowels (A, I, O) in the 3 even positions (2, 4, 6). The number of ways to arrange 3 vowels in 3 positions is given by the factorial of the number of vowels:

\[ \text{Ways to arrange vowels} = 3! = 6 \]

### Step 4: Place the consonants in the odd positions
The odd positions available are 1, 3, 5, and 7. We have 4 consonants (R, N, B, W) to place in these 4 positions. The number of ways to arrange 4 consonants in 4 positions is given by the factorial of the number of consonants:

\[ \text{Ways to arrange consonants} = 4! = 24 \]

### Step 5: Calculate the total arrangements
To find the total number of arrangements of the letters in "RAINBOW" with the specified conditions, we multiply the number of arrangements of vowels by the number of arrangements of consonants:

\[ \text{Total arrangements} = \text{Ways to arrange vowels} \times \text{Ways to arrange consonants} = 6 \times 24 = 144 \]

### Final Answer
The total number of words that can be formed such that the vowels occupy only even positions is **144**.

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