# Counting 7-Letter Palindromes: 26^7 Possibilities

• SammC
In summary, there is a problem statement about determining the number of possible seven-letter palindromes that can be made from the 26 letters in the English alphabet. The conversation discusses two possible approaches to solving this problem, one involving subtracting non-palindromic strings and the other considering different cases of letters in the palindrome. Ultimately, the solution involves starting with the basic form of a seven-letter palindrome as ABCDCBA and calculating the number of possibilities from there. The correct answer is 26^4, as some possibilities would be overcounted if calculated as 26*25*24*23.
SammC
The problem statement
There are 26 letters in the English Alphabet, how many seven-letter palindromes can be made?

The attempt at a solution
There are 26 letters in the alphabet, so there are 26^7 possible
strings of length 7 (order being important for palindromes, i don't
think 26 choose 7 is appropriate).

One way to do this would be to subtract the number of strings that are not palindromes from 26^7, but I have no idea how to get this number.

Another way to do it is to figure out how many palindromes
match the following cases:

7 of the same letter: 26 cases
6 of the same letter: 26*25 cases
5 of the same letter: ? cases
4 of the same letter: (ex: XXYZYXX)
3 of the same letter: (ex: YZXXXZY)
2 of the same letter: (ex: ZYXWXYZ)

Since after the first two cases, there are multiple ways to arrange
all of the letters that work, i get confused. (for example, 5 can be
arranged as XXYXYXX, or XYXXXYX, or YXXXXXY)

I know if I add all the cases together, i'll get the correct answer,
(subtracting overlap), but this gets out of hand very quickly. Is
there another approach that will work?

You could start by noting that a seven letter palindrome should look like
ABCDCBA
where A, B, C, D are any letter from the English alphabet.

Ah, I see.

So you have 26 choices for A, 25 choices for B, 24 choices for C, and 23 choices for D.

26*25*24*23 = 358800?

EDIT: Actually, the above is incorrect, A, B, C , and D can all be the same.

is 26^4 correct? That seems like it would be over counting, or counting some possibilities multiple times?

I would say 26^4, yes.
Which two words, for example, would be overcounted then?

## What is a palindrome?

A palindrome is a word, phrase, or sequence that reads the same backward as forward. For example, "radar" and "madam" are both palindromes.

## How many 7-letter palindromes are there?

There are a total of 26^7 (approximately 8.03 million) possible 7-letter palindromes using the 26 letters of the English alphabet.

## What is the significance of 7 letters?

7 is the number of letters that can be used to create the most unique palindromes using the 26 letters of the alphabet. Going above 7 letters would result in duplicate palindromes.

## Is there a formula for calculating the number of possible 7-letter palindromes?

Yes, the formula is 26^(n/2), where n is the number of letters in the palindrome. For example, a 5-letter palindrome would have 26^(5/2) (approximately 308 million) possibilities.

## Are all 7-letter palindromes actual words?

No, not all 7-letter palindromes are actual words. However, there are many 7-letter palindromic words such as "radar", "level", and "rotator".

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