2002 AIME II Problems/Problem 1

Problem

Given that

$\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\

   &(2)& \text{y is the number formed by reversing the digits of x; and}\\

&(3)& z=|x-y|. \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

How many distinct values of are possible?

Solution

We count the number of three-letter and three-digit palindromes, then subtract the number of license plates containing both types of palindrome.

There are letter palindromes, digit palindromes, and palindromes that contain both letters and digits.

Since there are possible plates, the probability desired is . Thus .