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 together?

Options

A. 720
B. 360
C. 540
D. 1440

Answer

To find the number of arrangements of the word "RAINBOW" where the letters R and W are together, we can follow these steps:

### Step 1: Treat R and W as a single unit
Since we want R and W to be together, we can treat them as a single entity or unit. Let's denote this unit as (RW).

### Step 2: Count the total units
Now, instead of 7 individual letters in "RAINBOW", we have the following units:
- (RW)
- A
- I
- N
- B
- O

This gives us a total of 6 units: (RW), A, I, N, B, O.

### Step 3: Calculate the arrangements of these units
The number of ways to arrange these 6 units is given by the factorial of the number of units:
\[ 6! = 720 \]

### Step 4: Consider the arrangements of R and W
Within the unit (RW), R and W can be arranged in 2 different ways: RW or WR.

### Step 5: Calculate the total arrangements
To find the total arrangements where R and W are together, we multiply the number of arrangements of the units by the arrangements of R and W:
\[ \text{Total arrangements} = 6! \times 2 = 720 \times 2 = 1440 \]

### Final Answer
Thus, the total number of arrangements of the word "RAINBOW" where R and W are together is **1440**.

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