Answer these questions independently of each other
In the different arrangements of the word RAINBOW, how many words are there in which R and W are never together?

Options

A. 3600
B. 2400
C. 1774
D. 1440

Answer

To solve the problem of finding the number of arrangements of the word "RAINBOW" in which the letters R and W are never together, we can follow these steps:

### Step 1: Calculate the total arrangements of the word "RAINBOW".
The word "RAINBOW" consists of 7 distinct letters. 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 R and W are together.
To find the arrangements where R and W are together, we can treat R and W as a single unit or block. This means we will consider the block (RW) as one letter.

Now, if we treat (RW) as one letter, we have the following letters to arrange: (RW), A, I, N, B, O. This gives us a total of 6 units to arrange.

\[ \text{Arrangements with R and W together} = 6! = 720 \]

### Step 3: Consider the arrangements of R and W within their block.
Since R and W can be arranged in 2 different ways (RW or WR), we need to multiply the arrangements from Step 2 by 2.

\[ \text{Arrangements with R and W together} = 6! \times 2 = 720 \times 2 = 1440 \]

### Step 4: Calculate the arrangements where R and W are never together.
To find the arrangements where R and W are never together, we subtract the number of arrangements where R and W are together from the total arrangements.

\[ \text{Arrangements where R and W are never together} = \text{Total arrangements} - \text{Arrangements with R and W together} \]

\[ = 5040 - 1440 = 3600 \]

### Final Answer:
The number of arrangements of the word "RAINBOW" in which R and W are never together is **3600**.

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