Answer these questions independently of each other
In the different arrangements of the word RAINBOW, how many words are there in which vowels occupy odd positions?

Options

A. 676
B. 625
C. 343
D. 576

Answer

To solve the problem of how many arrangements of the word "RAINBOW" have vowels occupying odd positions, we can follow these steps:

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

### Step 2: Determine the positions in the word
In a 7-letter word, the positions are:
1, 2, 3, 4, 5, 6, 7

The odd positions are:
1, 3, 5, 7 (4 odd positions)

### Step 3: Choose positions for the vowels
We need to place the 3 vowels (A, I, O) in the 4 available odd positions. We can choose 3 positions out of the 4 for the vowels. The number of ways to choose 3 positions from 4 is given by the combination formula:

\[ \text{Number of ways to choose positions} = \binom{4}{3} = 4 \]

### Step 4: Arrange the vowels in the chosen positions
Once we have chosen the 3 positions for the vowels, we can arrange the 3 vowels (A, I, O) in those positions. The number of arrangements of 3 vowels is given by:

\[ \text{Number of arrangements of vowels} = 3! = 6 \]

### Step 5: Arrange the consonants in the remaining positions
After placing the vowels, we have 4 positions left (1 odd position is left and 3 even positions). We need to arrange the 4 consonants (R, N, B, W) in these 4 positions. The number of arrangements of 4 consonants is given by:

\[ \text{Number of arrangements of consonants} = 4! = 24 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements where the vowels occupy odd positions by multiplying the number of ways to choose the positions, the arrangements of the vowels, and the arrangements of the consonants:

\[ \text{Total arrangements} = \binom{4}{3} \times 3! \times 4! = 4 \times 6 \times 24 \]

Calculating this gives:

\[ \text{Total arrangements} = 4 \times 6 \times 24 = 576 \]

### Final Answer
Thus, the total number of arrangements of the word "RAINBOW" in which the vowels occupy odd positions is **576**.

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