Problem 34 - Algebra and Trigonometry

Question

How many five-letter palindromes are possible? (A palindrome is a string of letters that reads the same backward and forward, such as the string XCZCX.)

Step-by-Step Solution

Answer

There are 17,576 possible five-letter palindromes.

Step 1: Understand the Structure of a Palindrome

A five-letter palindrome has a symmetry where the 1st and 5th letters are the same, the 2nd and 4th letters are the same, and the middle letter, the 3rd one, stands alone. So a string like 'ABCBA' forms a palindrome.

Step 2: Calculate Possible Choices for Letter Positions

Each of the three positions (1st, 2nd, and 3rd) in the palindrome can be filled with any of the 26 letters of the alphabet, since 5-letter palindromes repeat these letters in symmetric positions.

Step 3: Calculate Total Number of Palindromes

Since you have 26 choices for each of the three important positions, use multiplication: \( 26 \times 26 \times 26 \). This gives us the total number of five-letter palindromes.

Step 4: Perform the Calculation

Perform the multiplication: \( 26 \times 26 \times 26 = 17576 \). Thus, there are 17,576 possible five-letter palindromes.

Key Concepts

five-letter words

Palindromes are a fascinating type of five-letter word because they read the same both forward and backward. They follow a unique pattern of symmetry. In a five-letter palindrome, the 1st and 5th letters must be identical, as do the 2nd and 4th letters. This leaves the middle 3rd letter to stand on its own. An example of a five-letter palindrome is the word 'RADAR'. Here, 'R' is positioned as both the 1st and 5th letter, while 'A' repeats as the 2nd and 4th letter with 'D' in the center.

This unique requirement poses an interesting challenge in word formation. It leads to fewer possibilities than a random arrangement of letters, where each can vary without restriction. However, these constraints also allow us to systematically count all possible combinations without overlooking any options.

alphabet symmetry

The concept of symmetry in palindromes is essential. It is the harmony in the arrangement of letters that ensures the word reads the same from either direction. To stick with the palindrome structure:

• The 1st and 5th letters mirror each other.
• The 2nd and 4th letters are also identical.
• The 3rd letter is unique because it does not have a mirrored counterpart.

This symmetry is not just visually appealing but also simplifies the process of constructing a palindrome. By knowing that paired letters must be the same, we reduce complexity in letter choice.

Understanding this concept of alphabet symmetry makes it easier to grasp why certain letter arrangements work while others don't. It’s these mirrored pairs that help maintain the strict palindrome conditions throughout.

combinatorics

Combinatorics is the branch of mathematics that deals with counting combinations and arrangements efficiently. In the context of five-letter palindromes, it helps us calculate the number of possible palindrome words.

The method begins by choosing the number of potential variations for each letter position that impacts the palindrome's design:

• The 1st, 2nd, and 3rd positions can each be filled with any of the 26 letters from the English alphabet.
• Since positions 4 and 5 are duplicates of 2 and 1, respectively, we don't need to multiply for them separately.

Thus, to find the total number of variations possible for a five-letter palindrome, you use the formula: \( 26 \times 26 \times 26 \). This calculation gives us 17,576 potential palindromes. By applying combinatorics, we ensure a precise count of potential palindromic words, accounting for all constraints imposed by their structure.