A8. How many three-digit integers less than 1000 have exactly two different digits in their representation (for example, 232 , or 466 )?
SolUtion
There are 9 integers with two zeros, i.e. $100,200, \cdots, 800,900$. When the repeated digit is non-zero, the integers have the form ' $x x y$ ', ' $x y x$ ' or ' $y x x$ '. If $x=1, y$ can be $0,2,3,4,5,6,7,8$ or 9 , although we must ignore ' 011 ' as this is a two-digit integer. This gives 26 different integers. Similarly, there will be an additional 26 integers for every non-zero $x$ value. Therefore, the total number of three-digit integers less than 1000 that have exactly two different digits in their representation is $9+9 \times 26=243$.