In the question number 13 how many words are there in which vowels are always before the consonants ?

Options

A. 72
B. 144
C. 96
D. none of these

Answer

To solve the problem of how many words can be formed from the letters of the word "RAINBOW" such that all vowels appear before the consonants, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "RAINBOW" consists of the following letters:
- Vowels: A, I, O
- Consonants: R, N, B, W

### Step 2: Count the vowels and consonants
We have:
- 3 vowels (A, I, O)
- 4 consonants (R, N, B, W)

### Step 3: Arrange the vowels and consonants
Since the vowels must always come before the consonants, we can treat the arrangement as two separate groups:
1. Arrange the vowels (A, I, O)
2. Arrange the consonants (R, N, B, W)

### Step 4: Calculate the arrangements for vowels
The number of ways to arrange the 3 vowels is given by the factorial of the number of vowels:
- Arrangements of vowels = 3! = 3 × 2 × 1 = 6

### Step 5: Calculate the arrangements for consonants
The number of ways to arrange the 4 consonants is given by the factorial of the number of consonants:
- Arrangements of consonants = 4! = 4 × 3 × 2 × 1 = 24

### Step 6: Combine the arrangements
Since the arrangements of vowels and consonants are independent, we multiply the number of arrangements of vowels by the number of arrangements of consonants:
- Total arrangements = (Arrangements of vowels) × (Arrangements of consonants)
- Total arrangements = 3! × 4! = 6 × 24 = 144

### Final Answer
The total number of words in which the vowels are always before the consonants is **144**.

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