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?