In the question number 13 how many words are there in which N and B are together ?

Options

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

Answer

To solve the problem of how many words can be formed with the letters of the word "RAINBOW" such that the letters N and B are together, we can follow these steps:

### Step 1: Treat N and B as a single unit
Since we want N and B to be together, we can consider them as one single unit or block. Therefore, if we treat "NB" (or "BN") as one unit, we can represent the letters of "RAINBOW" as follows:

- Units: (NB), R, A, I, O, W

This gives us a total of 6 units to arrange: (NB), R, A, I, O, W.

### Step 2: Calculate the arrangements of the 6 units
Now, we need to find the number of ways to arrange these 6 units. The number of arrangements of n distinct objects is given by n!.

Thus, the number of arrangements of the 6 units is:
\[ 6! = 720 \]

### Step 3: Arrange N and B within their unit
Within the unit (NB), N and B can be arranged in 2 ways: either as "NB" or "BN".

### Step 4: Calculate the total arrangements
To find the total number of arrangements where N and B are together, we multiply the arrangements of the 6 units by the arrangements of N and B within their unit:
\[ \text{Total arrangements} = 6! \times 2! = 720 \times 2 = 1440 \]

### Final Answer
Therefore, the total number of words in which N and B are together is **1440**.

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