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Step 1: Understand the Problem
We need to form four-letter “words” (not necessarily meaningful) from the letters of "MEDITERRANEAN". The first letter must be R, and the fourth (last) letter must be E. We must count the number of distinct ways to form these words.
Step 2: Identify the Available Letters
The word “MEDITERRANEAN” has the following letters and their frequencies:
M (1)
E (3)
D (1)
I (1)
T (1)
R (2)
A (2)
N (2)
Since our four-letter word must start with R and end with E, this immediately uses up 1 R and 1 E. We need to fill the remaining two positions (the 2nd and 3rd letters) from the leftover pool of letters.
Step 3: Adjust the Letter Pool
After using 1 R out of 2, and 1 E out of 3, the remaining letters are:
M (1)
E (2)
D (1)
I (1)
T (1)
R (1)
A (2)
N (2)
Step 4: Count the Possible Fills for the 2nd and 3rd Positions
We have two cases:
Case A: The two positions are filled with the same letter
We look for letters that appear at least twice in the remaining pool: E, A, N, and R (note R has only 1 left, so it cannot fill both positions). Thus, the letters that can fill both vacant positions identically are:
E (2 available)
A (2 available)
N (2 available)
So there are 3 ways to fill the two positions with the same letter.
Case B: The two positions are filled with distinct letters
The different letters in the pool are M, E, D, I, T, R, A, N (8 distinct letters). To choose 2 distinct letters from these 8, we have $ ^8C_2 $ ways. Each choice of 2 distinct letters can be arranged in $2!$ ways in the two positions. Hence, the total number of ways for this case is:
$ ^8C_2 \times 2! $
Step 5: Calculate the Total Number of Words
The total number of valid four-letter words is the sum of the two counts (same letter case + distinct letters case):
$ \text{Total} = 3 + \left(^8C_2 \times 2!\right).
Compute $ ^8C_2 = \frac{8 \times 7}{2} = 28 $ and then $ 28 \times 2! = 56 $. Therefore,
$ \text{Total} = 3 + 56 = 59.
Answer
The total number of four-letter words of the form R _ _ E using letters from “MEDITERRANEAN” is 59.
