how many different words can be formed with the letters RAINBOW so that the vowels occupy odd places ?

Options

A. a)676
B. b)625
C. c)343
D. d)576

Answer

To solve the problem of how many different words can be formed with the letters of "RAINBOW" such that the vowels occupy odd places, we can follow these steps:

### Step 1: Identify the letters and their types
The letters in "RAINBOW" are:
- Vowels: A, I, O (3 vowels)
- Consonants: R, N, B, W (4 consonants)

### Step 2: Determine the positions available
The word "RAINBOW" has 7 letters, which means we have 7 positions:
1. 1st position (odd)
2. 2nd position (even)
3. 3rd position (odd)
4. 4th position (even)
5. 5th position (odd)
6. 6th position (even)
7. 7th position (odd)

The odd positions are 1, 3, 5, and 7 (total of 4 odd positions).

### Step 3: Place the vowels in the odd positions
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
Once we have chosen the 3 positions, we can arrange the 3 vowels in those positions. The number of arrangements of 3 vowels is given by:
\[ \text{Arrangements of vowels} = 3! = 6 \]

### Step 5: Place the consonants in the remaining positions
After placing the vowels, we have 4 consonants (R, N, B, W) that need to be placed in the remaining 4 positions (1 odd position left and 2 even positions). The number of arrangements of 4 consonants is given by:
\[ \text{Arrangements of consonants} = 4! = 24 \]

### Step 6: Calculate the total arrangements
Now, we multiply the number of ways to choose the vowel 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:
\[ 4 \times 6 = 24 \]
\[ 24 \times 24 = 576 \]

### Final Answer
Thus, the total number of different words that can be formed with the letters of "RAINBOW" such that the vowels occupy odd places is **576**.

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