How many different 4 letter words (not necessarily meaningful) can be formed using the letters of the word MEDITERRANEAN such that the first letter is E and the last letter is R?

Introduction / Context:
This question combines permutations with repeated letters. The word MEDITERRANEAN has several repeated letters, and we restrict the first and last positions of the 4 letter word. We must count distinct arrangements respecting letter multiplicities and position constraints, a typical challenge in combinatorics.

Given Data / Assumptions:

• The base word is MEDITERRANEAN.
• Letter counts are: E 3 times, R 2 times, A 2 times, N 2 times, and M, D, I, T once each.
• We need 4 letter words with first letter fixed as E and last letter fixed as R.
• Letters cannot be used more times than they appear in the original word.
• Words are considered distinct based on letter sequence; they need not be meaningful.

Concept / Approach:
Because the first and last letters are fixed, we only need to choose and arrange the two middle letters. We treat MEDITERRANEAN as a multiset of available letters. After using one E and one R for the first and last positions, we find how many distinct 2 letter sequences can be formed from the remaining letters, respecting their remaining counts. This is a multiset permutation problem of length 2.

Step-by-Step Solution:
Step 1: Count letters in MEDITERRANEAN. We have E 3, R 2, A 2, N 2 and M, D, I, T 1 each.Step 2: Fix the first letter as E and the last letter as R. After using one E and one R, the remaining counts are: E 2, R 1, A 2, N 2, M 1, D 1, I 1, T 1.Step 3: We must fill the second and third positions using these remaining letters.Step 4: Consider the 8 distinct letter types {E, R, A, N, M, D, I, T}. We count all possible ordered pairs that respect multiplicities.Step 5: If the two middle letters are different, any ordered pair of two different letters from the 8 types is allowed, since only one of each type is used. Number of such ordered pairs is 8 * 7 = 56.Step 6: If the two middle letters are the same, we need letters with at least 2 copies remaining. These are E (2 left), A (2) and N (2), contributing middle pairs EE, AA and NN. That gives 3 additional words.Step 7: Total distinct 4 letter words = 56 + 3 = 59.

Verification / Alternative check:

Why Other Options Are Wrong:

• 56: This omits the cases where the two middle letters are the same and counts only distinct letter pairs.
• 64 and 55: These values do not correspond to any correct splitting into identical and distinct pairs and likely come from miscounting the multiplicity of E, A and N.

Common Pitfalls:
Students may treat all remaining letters as distinct and simply compute 8^2 = 64, ignoring the upper bounds from the original word. Others count only combinations without considering order, which would undercount. A frequent oversight is forgetting that E, A and N can each appear twice in the middle positions because they still have at least 2 copies left.

Final Answer:
The number of different 4 letter words with first letter E and last letter R is 59.