Counting 7-Letter Palindromes: 26^7 Possibilities

[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?

 

[CompuChip]

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.

 

[SammC]

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?

 

[CompuChip]

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