In how many ways the letters of the words GARGANTUNG be rearranged such that all the G's appear together?

Options

A. `(8!)/(3!cdot2!cdot2!)`
B. `(8!)/(2!cdot2!)`
C. `(8!cdot3!)/(2!cdot2!)`
D. `(8!)/(2!cdot3!)`

Answer

To solve the problem of rearranging the letters of the word "GARGANTUNG" such that all the G's appear together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Letters**: The word "GARGANTUNG" consists of the letters G, A, R, G, A, N, T, U, N, G. We can see that there are 3 G's, 3 A's, 2 N's, 1 R, and 1 T.

2. **Group the G's Together**: Since we want all the G's to appear together, we can treat the three G's as a single unit or block. This means we can represent the block of G's as [GGG].

3. **List the Remaining Letters**: After grouping the G's, the remaining letters are A, R, A, N, T, U, N. So, we now have the following units to arrange: [GGG], A, R, A, N, T, U, N.

4. **Count the Total Units**: This gives us a total of 8 units to arrange: [GGG], A, R, A, N, T, U, N.

5. **Calculate Arrangements**: The total number of arrangements of these 8 units can be calculated using the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]
where \( n \) is the total number of items to arrange, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating items.

6. **Substituting Values**: In our case, we have:
- Total units (n) = 8
- A appears 2 times
- N appears 2 times

Therefore, the number of arrangements is:
\[ \text{Number of arrangements} = \frac{8!}{2! \times 2!} \]

7. **Calculating Factorials**:
- \( 8! = 40320 \)
- \( 2! = 2 \)

So, we calculate:
\[ \text{Number of arrangements} = \frac{40320}{2 \times 2} = \frac{40320}{4} = 10080 \]

8. **Final Answer**: Thus, the total number of ways to rearrange the letters of the word "GARGANTUNG" such that all the G's appear together is **10080**.