Problem:
Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is , where and are relatively prime positive integers. Find .
Solution:
The probability that a license plate will have a three-letter palindrome is because there are possibilities for each of the first two letters, and the third letter must be the same as the first. Similarly, the probability that a license plate will have a three-digit palindrome is . The probability that a license plate will have both a three-letter palindrome and a three-digit palindrome is . Apply the Inclusion-Exclusion Principle to conclude that the probability that a license plate will have at least one palindrome is
Thus .
The problems on this page are the property of the MAA's American Mathematics Competitions