To solve the problem of arranging the letters of the word "GOSSAMERE" such that all S's and M's appear in the middle, we can follow these steps:
### Step 1: Identify the letters and their counts
The word "GOSSAMERE" consists of the following letters:
- G: 1
- O: 1
- S: 2
- A: 1
- M: 1
- E: 2
- R: 1
### Step 2: Determine the positions for S's and M's
Since we want all S's and M's to be in the middle, we can treat the S's and M's as a block. The arrangement of S's and M's will be in the form of "SSM" or "SMS" or "MSS".
### Step 3: Arrange the remaining letters
After placing the S's and M's in the middle, we have the remaining letters: G, O, A, E, E, R.
### Step 4: Count the arrangements of the remaining letters
The remaining letters are G, O, A, E, E, R, which consist of 6 letters where E repeats twice. The number of arrangements of these letters can be calculated using the formula for permutations of multiset:
\[
\text{Arrangements} = \frac{n!}{p_1! \cdot p_2! \cdots}
\]
Where \( n \) is the total number of letters, and \( p_1, p_2, \ldots \) are the counts of each repeated letter.
For our case:
- Total letters = 6 (G, O, A, E, E, R)
- E repeats 2 times.
Thus, the arrangements of the remaining letters are:
\[
\text{Arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360
\]
### Step 5: Count the arrangements of S's and M's
The S's and M's can be arranged in the middle in the following ways:
- S, S, M can be arranged as:
\[
\text{Arrangements} = \frac{3!}{2!} = \frac{6}{2} = 3
\]
### Step 6: Calculate the total arrangements
Now, to find the total arrangements, we multiply the arrangements of the remaining letters by the arrangements of S's and M's:
\[
\text{Total Arrangements} = \text{Arrangements of remaining letters} \times \text{Arrangements of S's and M's}
\]
\[
\text{Total Arrangements} = 360 \times 3 = 1080
\]
### Final Answer
Thus, the total number of ways to arrange the letters of the word "GOSSAMERE" such that all S's and M's appear in the middle is **1080**.
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