A five-letter word such as 'ROTOR' reads the same forwards and backwards and is called a palindrome. Using the 26 letters of the English alphabet, what is the maximum possible number of distinct five-letter palindromes that can be formed?

Introduction / Context:
This problem concerns palindromic words, which read the same from left to right and right to left. A five-letter palindrome has a specific mirrored structure, which greatly restricts how letters can be chosen compared to arbitrary five-letter words. The question asks for the maximum number of five-letter palindromes that can be formed using the 26 letters of the English alphabet, allowing any letters to be repeated, as long as the palindromic structure is obeyed.

Given Data / Assumptions:
- We have 26 possible letters for each position (the English alphabet).

- We are forming five-letter strings.

- Each string must satisfy the palindrome property: it reads the same forwards and backwards.

- Letters may be repeated; there is no restriction that all letters must be distinct.

Concept / Approach:
A five-letter palindrome has the form a b c b a when written from left to right. The first and fifth letters must match, the second and fourth letters must match, and the middle letter can be any letter. Thus, although there are five positions, only three of them are independent: the first, second, and third positions. Once these three are chosen, the last two positions are determined automatically by the palindrome requirement. Therefore, the total number of five-letter palindromes equals the number of ways to choose letters for these three independent positions.

Step-by-Step Solution:
Step 1: Consider the general structure of a five-letter palindrome: positions 1, 2, 3, 4, 5 must form the pattern a b c b a.Step 2: The letter in position 1 can be any of the 26 letters.Step 3: Once position 1 is chosen, position 5 must be the same letter, and thus has no further independent choice.Step 4: The letter in position 2 can also be any of the 26 letters.Step 5: Once position 2 is chosen, position 4 must match it and is therefore determined.Step 6: The letter in position 3 (the middle letter) is completely free and can again be any of the 26 letters.Step 7: Therefore, the total number of five-letter palindromes is 26 * 26 * 26 = 26^3.Step 8: Compute 26^3 = 26 * 26 * 26 = 676 * 26 = 17576.

Verification / Alternative check:
You can think of the process as choosing a triple (a, b, c) where each of a, b and c is any letter from the 26-letter alphabet. For each such triple, there is exactly one corresponding palindrome a b c b a. This creates a one-to-one correspondence between the set of all triples and the set of all palindromes, so counting triples directly gives the count of palindromes. Since there are 26 choices for each of the three letters, this again yields 26^3 = 17576.

Why Other Options Are Wrong:
- 17756 is close but not equal to 26^3, and there is no natural combinatorial structure giving exactly this number for five-letter palindromes.

- 12657 and 12666 are also not equal to any simple power or product of 26 associated with independent positions in a five-letter palindrome and therefore do not match the correct structural reasoning.

Common Pitfalls:
Some learners mistakenly think that all five positions are independent and compute 26^5, which counts all five-letter strings, not just palindromes. Others may incorrectly enforce extra conditions, such as no repeated letters, which is not stated in the problem. The key is recognising that palindromes greatly reduce independence among positions and that only the first three letters need to be freely chosen.

Final Answer:
The maximum possible number of distinct five-letter palindromes that can be formed using the 26 letters of the English alphabet is 17576.