A teacher prepares a test

This is a combination question, since the order in which the questions are answered is not important.

All students have to answer 4 questions out of 5, which means that students can skip exactly one question, any single question.

So there are two cases:
Either
1. the student answers 1 question out of the first 2, and 3 questions out the the last 3,
or
2. the student answers 2 questions out of the first 2 and 2 questions out of the last 3.

Case 1:
There are 3 choices in question 1 or 2, 4 choices each in the remaining 3 questions, so the number of answers for the particular choice of questions is 3*4*4*4=192.
The number of choices of questions is
(2,1)=2 for the first group, and (3,3)=1
for a total of 2*1=2 choices of questions for case 1.
The total number of answers for case can be found by multiplying the number of choice of questions with the number of possible answers for each selection of questions = 2*192=384.

Case 2:
Number of possible answers for each selected combination of questions
= 3*3*4*4 = 144
Number of possible choice of questions:
(2,2)*(3,2)=1*3=3
Total number of possible answers for case 2 = 144*3=432

Total possible answers = 384+432 = 816