IF no two consonants are together then in how many ways can the word RAINBOW be arranged ?

Options

A. 441
B. 420
C. 360
D. 144

Answer

To solve the problem of arranging the letters in the word "RAINBOW" such that no two consonants are together, we can follow these steps:

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

### Step 2: Arrange the vowels
First, we will arrange the vowels. The vowels A, I, O can be arranged in different ways. The number of arrangements of 3 vowels is given by the factorial of the number of vowels.

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

### Step 3: Create slots for consonants
Once the vowels are arranged, we can create slots for the consonants. When we arrange the 3 vowels, we can visualize them as follows:

- Vowel arrangement: _ V _ V _ V _

This arrangement creates 4 slots (indicated by underscores) where the consonants can be placed.

### Step 4: Place the consonants in the slots
Now, we need to place the 4 consonants R, N, B, W in these 4 slots. Since we have 4 consonants and 4 slots, we can arrange the consonants in these slots in different ways. The number of arrangements of 4 consonants is given by the factorial of the number of consonants.

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

### Step 5: Calculate the total arrangements
To find the total arrangements of the word "RAINBOW" such that no two consonants are together, we multiply the number of arrangements of vowels by the number of arrangements of consonants.

\[ \text{Total arrangements} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 6 \times 24 = 144 \]

Thus, the total number of ways to arrange the letters in the word "RAINBOW" such that no two consonants are together is **144**.

### Final Answer:
The required answer is **144**.

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