At a fund-raiser, students pay $5 for the chance to throw a pie in the faces of their teachers. The student flips a coin twice. If the results are the same both times (two heads or two tails), the student's chosen teacher flips the same coin three times. If the teacher does not get the same results all three times, the student gets to hit the teacher in the face with a pie. However, if the student does not get the same results on both of his flips, and the teacher does get the same results on his three flips (three heads or three tails), the teacher gets to hit the student with a pie.
1) Calculate the probability that the student gets to hit the teacher.
2) Calculate the probability that the teacher gets to hit the student.
3) Calculate the probability that no one gets hit.

First calculate the probabilities of each coin flip scenario:

P(student's flips both the same)
P(student's flips not both the same)
P(teacher's flips all three the same)
P(teacher's flips not all three the same)

Note that you really only need to calculate two of those because the other two are simple:

P(student's flips not both the same) = 1 - P(student's flips both the same)
P(teacher's flips not all the same) = 1 - P(teacher's flips all the same)

Then your questions boil down to some compound probabilities:

P(student gets to hit teacher) = P(student's flips both the same) * P(teacher's flips not all the same)
P(teacher gets to hit student) = P(student's flips not both the same) * P(teacher's flips all the same)

Part (c) is trivial once you know the other two probabilities:
P(nobody gets hit) = 1 - P(somebody gets hit) = 1 - P(teacher gets to hit student) - P(student gets to hit teacher)

So ultimately this problem comes down to two different calculations:
P(student's flips both the same)
P(teacher's flips all three the same)

Do you know how to calculate those two probabilities?