Expected value of $\overline{ABC} \times \overline{DEF}$ Here is a question from HMMT:
https://hmmt-archive.s3.amazonaws.com/tournaments/2013/nov/team/solutions.pdf

The digits $1$, $2$, $3$, $4$, $5$, $6$ are randomly chosen (without replacement) to form the three-digit numbers $M = \overline{ABC}$ and $N = \overline{DEF}$. For example, we could have $M = 413$ and $N = 256$. Find the expected value of $M \cdot N$.

Here's what I did. Each digit on average is going to be $(1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5$, so the expected value is $(100(3.5) + 10(3.5) + 3.5)^2 = 150932.25$. However, the answer at the link above is $143745$. What did I do wrong? Did I overcount something?

The correct answer is 143745.
$$
\frac{1}{6}\cdot\frac{1\cdot(2+3+4+5+6)+2\cdot(1+3+4+5+6)+...+6\cdot(1+2+3+4+5)}{5}\cdot111\cdot111
$$
Your answer is incorrect because you average the numbers evenly
