Answer these questions independently of each other
In the different arrangements of the word RAINBOW, how many words are there in which no two consonants are together ?

Options

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

Answer

To find the number of arrangements of the word "RAINBOW" such that no two consonants are together, we can follow these steps:

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

### Step 2: Arrange the vowels
First, we can arrange the vowels. The number of ways to arrange 3 vowels (A, I, O) is given by:
\[ 3! = 6 \]

### Step 3: Create slots for consonants
When we arrange the vowels, we create slots where the consonants can be placed. For example, if we arrange the vowels as A, I, O, the arrangement looks like this:
- _ A _ I _ O _

In this arrangement, there are 4 slots (indicated by underscores) where we can place the consonants (before A, between A and I, between I and O, and after O).

### Step 4: Place the consonants in the slots
Since we have 4 consonants (R, N, B, W) and 4 slots, we can place each consonant in one of these slots. The number of ways to choose 4 slots from 4 available slots is:
\[ 4C4 = 1 \]
This means we have only one way to choose all slots since we have exactly 4 consonants and 4 slots.

### Step 5: Arrange the consonants
The number of ways to arrange the 4 consonants (R, N, B, W) is given by:
\[ 4! = 24 \]

### Step 6: Calculate the total arrangements
Now, we multiply the number of arrangements of vowels by the number of arrangements of consonants:
\[ \text{Total arrangements} = \text{(arrangements of vowels)} \times \text{(arrangements of consonants)} = 6 \times 24 = 144 \]

### Final Answer
The total number of arrangements of the word "RAINBOW" such that no two consonants are together is **144**.

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