Question

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A sequence of characters is called a palindrome if it reads the same way forward or backward. For example, 59AA95 is a six-character palindrome, and 59A95 is a five-character palindrome. Some other instances of palindromes: U NU, LON NOL, MALAYALAM, NOW ON, PUT UP, TOO HOT TO HOOT, NEVER ODD OR EVEN, ABLE WAS I ERE I SAW ELBA, and POOR DAN IS IN A DROOP. Find the number of nine-character palindromes that can be formed using the letters of the alphabet such that no letter appears more than twice in each of them.

Answer 1

Since there are odd number of letters in a 9. Character palindromes , any letter not occurring at the middle position will appear atleast twice (by the property of Palindrome) MALAYALAM Y is in the middle , while A's, l's and M's are Ocuving atleast twice. This means for a 9-character palindrome, formed by using any of 26-letters of alphabet, We have 26 options to fill the middle place Now , since we have to make a palindrome in which no letter appears more than twice, thus we will choose distinct letters from the remaining 25 letters of alphabet. We don't need to fill the remaining 8 positions, filling only the 4 positions either to the left on the right to the middle will automatically serve the purpose on the other side ( by the property of Palindrome) o, our task reduces to arranging 4 out of 25 distinct characters which can be done in 25 Py ways Also to fill middle place , number of ways = 26. Thus, total number of palindromes formed by using letter Il collength 9) of alphabet so that none of the letters occur more than twice . 26 x 25 Py = 7,893,600