Five 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1 or a 2?
There are $\binom{5}{2}=10$ ways to choose which of the five dice show a 1 or 2. The probability of any one of these occurring is $\left(\frac{1}{3}\right)^{\!2}\left(\frac{2}{3}\right)^{\!3}$. So the overall probability is $$10\left(\frac{1}{3}\right)^{\!2}\left(\frac{2}{3}\right)^{\!3}=\frac{10\times 2^3}{3^5}=\boxed{\frac{80}{243}}.$$