Palindromes
Probability
Level
3
Find the number of palindromes between 11 and (both inclusive).
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We need to find all palindromes with at most 10 digits.
1) Finding palindromes containing even digits - Pick any number between 1 and 99999, and append it mirrored, for example: 132.321 = 132231. You get a palindrome. So there are 99999 palindromes of even length smaller than 10^10.
2) Finding palindromes containing odd digits -
Now, if the length is odd, the middle digit is not important, so there are 10 times as many 2n+1 digit palindromes, as there are 2n digit palindromes. In other words, pick a number between 1 and 9999 and append it backwards keeping one position empty between our original and backwards number. Any number from 0 to 9 can fill that position. So there are 9999 ways to pick a number to append backwards times 10 ways to fill the empty gap, giving us 99990 palindromes of odd length less than 10^10. Eg-if you pick 123, palindrome obtained is 123x321 where x can take any number from 0-9.
So there are 99999 + 99990 = 199989 palindromes.