Problem 336

Each student answered exactly $2$ questions correctly, meaning that each
student answered exactly 1 question incorrectly.

Suppose that the correct answer to Question #3 is not $24$, and so each student answered Question
#3 incorrectly.

In this case, each student must have answered Question #2 correctly
(since each answered exactly $1$
question incorrectly). However, each student answered Question #2
differently, and so each student cannot have answered Question #2
correctly. This tells us that the correct answer to Question #3 is $24$, and each student answered this
question correctly.

Next, suppose that the correct answer to Question #1 is not $15$. In this case, Students A and C did
not answer Question #1 correctly, and so they both must have answered
Question #2 correctly (since each answered exactly 1 question
incorrectly). However, Students A and C have different answers for
Question #2 and thus both cannot have answered Question #2 correctly.
This tells us that the correct answer to Question #1 is $15$, and so Student B answered Question #1
incorrectly.

Finally, since Student B answered Question #1 incorrectly, then they
must have answered Question #2 correctly, and so the correct answer to
Question #2 is $38$.

The sum of the correct answers to the $3$ questions is $15+38+24=77$.

Question #1
Question #2
Question #3

Student A
$~~~~15~\checkmark$
$36$
$~~~~24~\checkmark$

Student B
$20$
$~~~~38~\checkmark$
$~~~~24~\checkmark$

Student C
$~~~~15~\checkmark$
$54$
$~~~~24~\checkmark$