In the word RAINBOW how many words are there in which vowels are never together ?

Options

A. 720
B. 1440
C. 360
D. 3600

Answer

To find the number of arrangements of the word "RAINBOW" such that the vowels are never together, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "RAINBOW" consists of 7 letters:
- Vowels: A, I, O
- Consonants: R, N, B, W

### Step 2: Calculate the total arrangements of the letters
First, we calculate the total arrangements of the letters in "RAINBOW". Since all letters are distinct, we can use the factorial of the total number of letters:
\[ \text{Total arrangements} = 7! = 5040 \]

### Step 3: Calculate arrangements where vowels are together
To find the arrangements where the vowels are together, we can treat the vowels A, I, O as a single unit or block. Thus, we can consider the block (AIO) along with the consonants R, N, B, W.

Now, we have the following units to arrange:
- Units: (AIO), R, N, B, W
- Total units = 5 (1 vowel block + 4 consonants)

The arrangements of these 5 units can be calculated as:
\[ \text{Arrangements of units} = 5! = 120 \]

Within the vowel block (AIO), the vowels can be arranged among themselves in:
\[ \text{Arrangements of vowels} = 3! = 6 \]

Thus, the total arrangements where vowels are together is:
\[ \text{Total arrangements with vowels together} = 5! \times 3! = 120 \times 6 = 720 \]

### Step 4: Calculate arrangements where vowels are never together
To find the arrangements where the vowels are never together, we subtract the arrangements where the vowels are together from the total arrangements:
\[ \text{Arrangements where vowels are never together} = 7! - (5! \times 3!) \]
\[ = 5040 - 720 = 4320 \]

### Final Answer
Thus, the number of arrangements of the word "RAINBOW" in which the vowels are never together is:
\[ \boxed{4320} \]