Adding all repeating decimals 0.abc with distinct digits

Consider every number of the form $0.\overline{abc}=0.abcabcabc\ldots$ where $a$, $b$, $c$ are three different digits from $0$ to $9$. What is the total of all such numbers?

360. There are $10\cdot9\cdot8=720$ choices of $(a,b,c)$. By symmetry each decimal place averages $4.5$, so the average number is $4.5\sum_{k\ge1}10^{-k}=\tfrac12$, and the total is $720\cdot\tfrac12=360$.