Different words are formed with the help of letters of the word SIGNATURE. Find the number of words in which
letters I, G and N are always together

Options

A. `8! xx 2!`
B. `7! xx 3!`
C. `6! xx 4!`
D. None of these

Answer

To solve the problem of finding the number of different words that can be formed using the letters of the word "SIGNATURE" where the letters I, G, and N are always together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Letters**: The word "SIGNATURE" consists of the letters S, I, G, N, A, T, U, R, E. There are a total of 9 letters.

2. **Group the Letters I, G, and N**: Since we want the letters I, G, and N to always be together, we can treat them as a single unit or block. Let's denote this block as [IGN].

3. **Count the Remaining Letters**: After grouping I, G, and N together, we have the following letters to arrange:
- The block [IGN]
- S, A, T, U, R, E

This gives us a total of 7 units to arrange: [IGN], S, A, T, U, R, E.

4. **Calculate the Arrangements of the Units**: The number of ways to arrange these 7 units is given by 7! (7 factorial).

5. **Arrange the Letters within the Block**: The letters I, G, and N can be arranged among themselves in 3! (3 factorial) ways.

6. **Combine the Results**: The total number of arrangements where I, G, and N are together is the product of the arrangements of the 7 units and the arrangements of the letters within the block:
\[ \text{Total Arrangements} = 7! \times 3! \]

7. **Calculate the Factorials**:
- \(7! = 5040\)
- \(3! = 6\)

Therefore,
\[ \text{Total Arrangements} = 5040 \times 6 = 30240 \]

### Final Answer:
The total number of different words that can be formed with the letters of the word "SIGNATURE" where I, G, and N are always together is **30,240**.