2016 AMC 10B Problem 6

To minimize the sum of the addends, use the two smallest nonzero hundreds digits, $1$ and $2$, and then the four smallest remaining digits, $0,3,4,5$. Placing the smaller digits in the tens places shows that every possible sum is at least $104+235=339$. On the other hand, any three-digit number with digit sum less than $4$ is at most $300$. Therefore the digit sum of $S$ is at least $4$. The example $157+243=400$ uses six distinct digits and has digit sum $4$. Thus the smallest possible digit sum is $4$. Thus, the correct answer is B.