Question

A professor teaching a discrete math course gives a multiple-choice quiz that has ten questions, each with four possible responses: a, b, c,
d. what is the minimum number of students that must be in the professor's class in order to guarantee that at least three answer sheets must be identical? (assume that no answers are left blank.)3

Answer 1

Each question has three possible distinct answers.
Ten questions have 3^10 possible distinct answers sheets.
=>
(3^10)+3 will guarantee that at least 3 answer sheets are identical.

Numerically, 
(3^10)+3 = 59049 +3 = 59052

Answer 2

(3^10)+3= 59049 + 3= 59052