A student takes a 5 question quiz with 4 choices for each question if the student guesses at random on each question what is the probability that the student gets exactly 4 questions correct?

This is abinomial distribution; number of trials (n) is 5, probability of success (p) is 1/4 or 0.25. With this information you can go to a Binomial Distribution Table and find the solution. Within the section of values for n=5 and p=.25, read from the section the probability of 4 which is 0.0146 (see related link for table).