Finding basic probabilities of events Mike is sitting a multiple-choice exam, each question has four answer options, only one is correct. Mike takes $5$ guesses.

a) What is the expected number of correct answers Mike obtains from the 5 guesses?

I put $\frac54 = 1.25$, is this right?

b) What is the probability that all 5 of his guesses are incorrect?

I put $(\frac34)^5 = \frac{243}{1024}$, is this right?

c) What is the probability exactly $4$ or the $5$ are correct?

I put $(\frac14)^4 \times \frac34 = \frac{3}{1024}$, is this correct?
Everything is fine except for your last calculation. Think about how many different ways you can have 4 out of 5 correct, and multiply that by the probability that 4 out of 5 are correct after you have decided which 4. (Hint: It should have both $1/4$ and $3/4$ in the formula)
Yes. Your solution in the comment, $(\frac{1}{4})^{4}\cdot\frac{3}{4}$, is the correct answer on the last question.