In the question number 13 How many words are there in which I and O are never together ?

Options

A. 1440
B. 720
C. 3600
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 the letters 'I' and 'O' are never together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "RAINBOW".
The word "RAINBOW" consists of 7 distinct letters: R, A, I, N, B, O, W. The total number of arrangements of these letters can be calculated using the factorial of the number of letters.

\[ \text{Total arrangements} = 7! = 5040 \]

### Step 2: Calculate the arrangements where 'I' and 'O' are together.
To find the arrangements where 'I' and 'O' are together, we can treat 'I' and 'O' as a single unit or block. This means we now have the following units to arrange: (IO), R, A, N, B, W. This gives us a total of 6 units to arrange.

\[ \text{Arrangements with IO together} = 6! = 720 \]

Since 'I' and 'O' can be arranged in 2 ways (IO or OI), we multiply the arrangements by 2.

\[ \text{Total arrangements with I and O together} = 6! \times 2 = 720 \times 2 = 1440 \]

### Step 3: Calculate the arrangements where 'I' and 'O' are never together.
To find the arrangements where 'I' and 'O' are never together, we can subtract the arrangements where they are together from the total arrangements.

\[ \text{Arrangements with I and O never together} = \text{Total arrangements} - \text{Arrangements with I and O together} \]

Substituting the values we calculated:

\[ \text{Arrangements with I and O never together} = 5040 - 1440 = 3600 \]

### Final Answer:
The number of arrangements of the letters in "RAINBOW" such that 'I' and 'O' are never together is **3600**.

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