Question:
We roll a fair 6-sided die 5 times.  What is the probability that exactly 3 of the 5 rolls are either a 1 or a 2?

Answer:
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 $\binom{5}{3}$ 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 \[\frac{\binom{5}{3}2^34^2}{6^5}=\boxed{\frac{40}{243}}.\]