How many 5-letter palindromes are there? This question may seem simple at first glance, but it actually requires a deeper understanding of the properties of palindromes and the limitations of the English alphabet. In this article, we will explore the number of 5-letter palindromes and the factors that contribute to their existence.
Palindromes are words, phrases, or numbers that read the same backward as forward. To create a 5-letter palindrome, we need to consider the structure of such a word. A 5-letter palindrome consists of two halves: the first half and the second half, which are mirror images of each other. The middle letter is unique, as it does not have a corresponding letter in the other half.
The English alphabet contains 26 letters. When creating a 5-letter palindrome, we have 26 options for the first letter, 26 options for the second letter, and 26 options for the third letter. However, the fourth letter must be the same as the second letter, and the fifth letter must be the same as the first letter. This reduces the number of possible combinations.
To calculate the number of 5-letter palindromes, we can use the following formula:
Number of 5-letter palindromes = (Number of options for the first letter) (Number of options for the second letter) (Number of options for the third letter)
In this case, the number of 5-letter palindromes is:
Number of 5-letter palindromes = 26 26 26 = 17,576
However, this calculation only takes into account palindromes with distinct letters. There are also palindromes with repeated letters, such as “aaaab” or “ababa.” To account for these, we need to consider the permutations of the letters within the palindrome.
For palindromes with repeated letters, we can use the following formula:
Number of palindromes with repeated letters = (Number of options for the first letter) (Number of options for the second letter) (Number of options for the third letter) (Number of options for the fourth letter)
In this case, the number of palindromes with repeated letters is:
Number of palindromes with repeated letters = 26 26 26 26 = 456,976
Therefore, the total number of 5-letter palindromes, including those with repeated letters, is:
Total number of 5-letter palindromes = Number of distinct palindromes + Number of palindromes with repeated letters
Total number of 5-letter palindromes = 17,576 + 456,976 = 474,552
In conclusion, there are 474,552 5-letter palindromes, including those with repeated letters. This number demonstrates the intricate relationship between the properties of palindromes and the limitations of the English alphabet.
