2002 AIME I Problems/Problem 1
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
Solution 1
Consider the three-digit arrangement, . There are choices for and choices for (since it is possible for ), and so the probability of picking the palindrome is . Similarly, there is a probability of picking the three-letter palindrome.
By the Principle of Inclusion-Exclusion, the total probability is
Solution 2
Using complementary counting, we count all of the license plates that do not have the desired property. In order to not be a palindrome, the first and third characters of each string must be different. Therefore, there are three digit non-palindromes, and there are three letter non palindromes. As there are total three-letter three-digit arrangements, the probability that a license plate does not have the desired property is . We subtract this from 1 to get as our probability. Therefore, our answer is .
See also
2002 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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