Number of ways in which the letters of the word `RAINBOW` be arranged such that `N` and `B` are together is

Options

A. a) `2560`
B. b) `1440`
C. c) `36`
D. d) `72`

Answer

To find the number of ways in which the letters of the word "RAINBOW" can be arranged such that 'N' and 'B' are together, we can follow these steps:

### Step 1: Treat 'N' and 'B' as a single unit
Since 'N' and 'B' need to be together, we can consider them as one single unit or block. Thus, we can represent this block as 'NB' or 'BN'.

### Step 2: Count the total units
The word "RAINBOW" has 7 letters: R, A, I, N, B, O, W. When we treat 'N' and 'B' as a single unit, we now have the following units to arrange:
- The block 'NB' (or 'BN')
- R
- A
- I
- O
- W

This gives us a total of 6 units to arrange: {NB/BN, R, A, I, O, W}.

### Step 3: Calculate the arrangements of the units
The number of ways to arrange these 6 units is given by the factorial of the number of units:
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

### Step 4: Consider the arrangements of 'N' and 'B'
Within the block, 'N' and 'B' can be arranged in 2 different ways: 'NB' or 'BN'. Therefore, we need to multiply the arrangements of the units by the arrangements of 'N' and 'B':
\[ 2! = 2 \]

### Step 5: Calculate the total arrangements
Now, we multiply the number of arrangements of the units by the arrangements of 'N' and 'B':
\[ \text{Total arrangements} = 6! \times 2! = 720 \times 2 = 1440 \]

### Final Answer
Thus, the number of ways in which the letters of the word "RAINBOW" can be arranged such that 'N' and 'B' are together is **1440**.