Palindrome Construction Explained

A palindrome is a word or phrase that reads the same forwards and backwards. For example, words like "LEVEL", "MADAM", and "REFER" are palindromes.

We are asked to find the number of different ways to construct a five-letter palindrome using the English alphabet. The English alphabet has 26 letters.

Let the five-letter palindrome be represented by the positions of its letters:

Position 1 | Position 2 | Position 3 | Position 4 | Position 5

For a five-letter word to be a palindrome, the letter at Position 1 must be the same as the letter at Position 5, and the letter at Position 2 must be the same as the letter at Position 4. The letter at Position 3 is the middle letter and stands alone.

So, the structure of a five-letter palindrome looks like this:

\(L_1 L_2 L_3 L_4 L_5\)

Where \(L_1 = L_5\) and \(L_2 = L_4\).

We need to determine the number of choices for each independent position:

• Position 1 (\(L_1\)): We can choose any of the 26 English alphabets for this position. So there are 26 options.
• Position 2 (\(L_2\)): We can choose any of the 26 English alphabets for this position. So there are 26 options.
• Position 3 (\(L_3\)): We can choose any of the 26 English alphabets for this position. So there are 26 options.
• Position 4 (\(L_4\)): Since this must be the same as \(L_2\), there is only 1 option once \(L_2\) is chosen.
• Position 5 (\(L_5\)): Since this must be the same as \(L_1\), there is only 1 option once \(L_1\) is chosen.

The total number of ways to construct such a palindrome is the product of the number of options for the independent choices (Positions 1, 2, and 3).

Calculating the Ways

Number of ways = (Options for Position 1) $\times$ (Options for Position 2) $\times$ (Options for Position 3)

Number of ways = \(26 \times 26 \times 26\)

This calculation represents the total number of possible five-letter palindromes that can be formed using the 26 English alphabets.

Let's look at the options provided:

• Option 1: \(26 \times 25 \times 24\)
• Option 2: \(26 \times 26\)
• Option 3: \(26 \times 26 \times 26\)
• Option 4: \(26 \times 26 \times 26 \times 26 \times 26\)

Comparing our result with the options, we find that the number of ways to construct a five-letter palindrome is \(26 \times 26 \times 26\).

This means there are \(26^3 = 17576\) different five-letter palindromes that can be constructed using English alphabets, regardless of whether they form meaningful words.