AMC 12 Mock Exam 2 Problem 2

Write the number as $\underline{a}\,\underline{b}\,\underline{c}$. It is divisible by $11$ exactly when $a-b+c$ is divisible by $11$. Since $a+b+c=11$, we have $a-b+c=11-2b$, which lies between $-7$ and $11$; it is a multiple of $11$ only if $11-2b=11$ (since $11-2b=0$ is impossible), i.e., $b=0$. Then $a+c=11$ with $1\le a\le9$ and $0\le c\le9$, so $a\in\{2,3,\dots,9\}$. There are $\boxed{8}$ such numbers: $209,308,\dots,902$.