### Instruction: Six 6-sided dice are rolled.  What is the probability that exactly two of the dice show a 1 and exactly two of the dice show a 2?  Express your answer as a common fraction.### Response: I know that the probability of rolling a 1 or a 2 on a single die is 1/3, and the probability of rolling something else is 2/3.
I also know that the order of the dice does not matter, so I can use combinations to count the number of favorable outcomes.
I need to choose two dice out of six to show a 1, which can be done in 6C2 = 15 ways.
Then, I need to choose two dice out of the remaining four to show a 2, which can be done in 4C2 = 6 ways.
The other two dice can show anything except 1 or 2, which can be done in 4^2 = 16 ways, since there are four possible outcomes for each die.
Therefore, the total number of favorable outcomes is 15 * 6 * 16 = 1440.
The total number of possible outcomes is 6^6 = 46656, since there are six possible outcomes for each die.
So, the probability of the event is 1440 / 46656, which can be simplified by dividing both numerator and denominator by 48.
The final answer is 30 / 972, which can be further simplified by dividing both numerator and denominator by 6.