Difference between revisions of "2002 AIME I Problems/Problem 1"
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== Solution == | == Solution == | ||
− | + | Consider the three-digit arrangement, <math>\overline{aba}</math>. There are <math>10</math> choices for <math>a</math> and <math>10</math> choices for <math>b</math> (since it is possible for <math>a=b</math>), and so the probability of picking the palindrome is <math>\frac{10 \times 10}{10^3} = \frac 1{10}</math>. Similarly, there is a <math>\frac 1{26}</math> probability of picking the three-letter palindrome. | |
− | <math>\frac{1}{26}+\frac{1}{10}-\frac{1}{260}=\frac{35}{260}=\frac{7}{52}\quad\Longrightarrow\quad7+52=\boxed{059}</math> | + | By the [[Principle of Inclusion-Exclusion]], the total probability is |
+ | <center><math>\frac{1}{26}+\frac{1}{10}-\frac{1}{260}=\frac{35}{260}=\frac{7}{52}\quad\Longrightarrow\quad7+52=\boxed{059}</math></center> | ||
== See also == | == See also == | ||
{{AIME box|year=2002|n=I|before=First Question|num-a=2}} | {{AIME box|year=2002|n=I|before=First Question|num-a=2}} | ||
+ | |||
+ | [[Category:Intermediate Combinatorics Problems]] | ||
+ | [[Category:Intermediate Probability Problems]] |
Revision as of 22:32, 24 April 2008
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
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
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 | ||
All AIME Problems and Solutions |