1. [15] Determine the number of palindromes that are less than 1000 .
Solution. Every one-digit number (there are nine) is a palindrome. The two-digit palindromes have the form $\underline{a} \underline{a}$ for a nonzero digit $a$, so there are nine of them. A three-digit palindrome is $\underline{a} \underline{b} \underline{a}$ with $a$ a nonzero digit and $b$ any digit, so there are $9 \times 10=90$ of these. Thus the number of palindromes less than 1000 is $9+9+90=\mathbf{1 0 8}$.