To solve the problem of forming 13-letter words from the letters of "MEDITERRANEAN" with the first letter as R and the fourth letter as E, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the letters in "MEDITERRANEAN":**
The letters in "MEDITERRANEAN" are:
- M, E, D, I, T, E, R, R, A, N, E, A, N
- The frequency of each letter is:
- M: 1
- E: 3
- D: 1
- I: 1
- T: 1
- R: 2
- A: 2
- N: 2
2. **Fix the positions:**
We need to fix the first letter as R and the fourth letter as E. Thus, the arrangement looks like this:
- R _ _ E _ _ _ _ _ _ _ _ _
3. **Count the remaining letters:**
After fixing R and E, we have the following letters left:
- M: 1
- E: 2 (since we used one E)
- D: 1
- I: 1
- T: 1
- R: 1 (since we used one R)
- A: 2
- N: 2
This gives us a total of 11 letters remaining.
4. **Determine the positions to fill:**
We need to fill the remaining 11 positions (2nd, 3rd, and 5th to 13th) with the remaining letters.
5. **Case Analysis:**
We will analyze two cases based on whether the letters in the 2nd and 3rd positions are the same or different.
**Case 1: Same letters in 2nd and 3rd positions**
- Possible pairs: (E, E), (A, A), (N, N)
- For each pair, the arrangement of the remaining letters can be calculated.
- Total arrangements for this case = 3 (one for each pair).
**Case 2: Different letters in 2nd and 3rd positions**
- We can choose 2 different letters from the remaining letters (M, E, D, I, T, R, A, N).
- The number of ways to choose 2 letters from 8 distinct letters is given by \( \binom{8}{2} \).
- Calculate \( \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28 \).
- Each pair of letters can be arranged in 2 ways (e.g., AB or BA).
- Total arrangements for this case = \( 28 \times 2 = 56 \).
6. **Combine both cases:**
- Total arrangements = Arrangements from Case 1 + Arrangements from Case 2
- Total arrangements = 3 + 56 = 59.
### Final Answer:
The total number of 13-letter words that can be formed is **59**.
