Answer these questions independently of each other
In the different arrangements of the word RAINBOW, how many words are there in which vowels are always before the consonants?

Options

A. (A) 72
B. (B) 144
C. (C) 96
D. (D) 124

Answer

To solve the problem of finding the number of arrangements of the word "RAINBOW" where the vowels always come 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 (3 vowels)
- Consonants: R, N, B, W (4 consonants)

### Step 2: Arrange the Vowels
Since the vowels must always come before the consonants, we first arrange the vowels. The number of ways to arrange the 3 vowels (A, I, O) is given by the factorial of the number of vowels:
\[ \text{Arrangements of Vowels} = 3! = 6 \]

### Step 3: Arrange the Consonants
Next, we arrange the consonants (R, N, B, W). The number of ways to arrange the 4 consonants is given by the factorial of the number of consonants:
\[ \text{Arrangements of Consonants} = 4! = 24 \]

### Step 4: Combine the Arrangements
Since the vowels must be arranged first followed by the consonants, the total number of arrangements where vowels are always before consonants is the product of the arrangements of vowels and consonants:
\[ \text{Total Arrangements} = \text{Arrangements of Vowels} \times \text{Arrangements of Consonants} = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer
Thus, the total number of arrangements of the word "RAINBOW" where the vowels are always before the consonants is **144**.

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