jmc

The number of possible rolls of 5 dice is $6^5$, since there are 6 possibilities for each of the 5 dice. Now we count the number of ways to get a 1 or a 2 in exactly 3 of the 5 rolls. First, we pick which 3 of the 5 rolls are 1 or 2: we can do that in $\left(\genfrac{}{}{0ex}{}{5}{3}\right)$ ways. Now for each of these 3 rolls, there are 2 choices, and for each of the other 2 rolls, there are 4 choices. Thus the probability is $$\frac{\left(\genfrac{}{}{0ex}{}{5}{3}\right)2^3{4}^2}{6^5}=\boxed{\frac{40}{243}}.$$