Key Concepts and Formulas:

• Frequency Analysis: Counting the occurrences of each distinct letter in the given word.
• Fundamental Counting Principle: If there are $m$ ways to do one thing, and $n$ ways to do another, then there are $m \times n$ ways to do both.
• Permutations: The number of ways to arrange $r$ distinct objects from a set of $n$ distinct objects is given by ${}^nP_r = \frac{n!}{(n-r)!}$.

Step-by-Step Solution:

1. Analyze the Letters in the Word "MEDITERRANEAN"

First, determine the frequency of each letter in the word "MEDITERRANEAN":

• M: 1
• E: 3
• D: 1
• I: 1
• T: 1
• R: 2
• A: 2
• N: 2

The word has a total of 13 letters.

2. Apply the Constraints

We want to form a four-letter word of the form R _ _ E. This means the first letter must be 'R' and the last letter must be 'E'.

3. Update Available Letters

Since we've used one 'R' and one 'E', we update the letter counts.

Original counts:

• M: 1
• E: 3
• D: 1
• I: 1
• T: 1
• R: 2
• A: 2
• N: 2

Updated counts:

• M: 1
• E: 2
• D: 1
• I: 1
• T: 1
• R: 1
• A: 2
• N: 2

4. Consider Possible Cases

We need to fill the two middle spaces. Since some letters have multiple occurrences, we divide into cases:

• Case 1: The two middle letters are the same.
• Case 2: The two middle letters are different.

5. Analyze Case 1: Identical Middle Letters

The letters that appear at least twice in the updated counts are E, A, and N. Therefore, the middle letters can be EE, AA, or NN. There are 3 possibilities in this case.

6. Analyze Case 2: Distinct Middle Letters

We have 8 distinct letters available (M, E, D, I, T, R, A, N). We need to choose 2 of them and arrange them. This is a permutation problem.

The number of ways to choose and arrange 2 distinct letters from 8 is ${}^8P_2 = \frac{8!}{(8-2)!} = \frac{8!}{6!} = 8 \times 7 = 56$.

7. Calculate the Total Number of Words

The total number of possible words is the sum of the number of words from each case:

Total = Case 1 + Case 2 = 3 + 56 = 59.

Common Mistakes & Tips:

• Remember to update the letter counts after placing the fixed letters.
• Distinguish between permutations and combinations. Since the order of the two middle letters matters, we use permutations.
• Be careful with casework when dealing with repeated letters.

Summary:

We analyzed the given word, applied the constraints to fix the first and last letters, and then considered two cases for filling the middle two letters: when they are identical and when they are distinct. Summing the counts from both cases gives the total number of possible words as 59.

Final Answer: The final answer is \boxed{59}, which corresponds to option (D).