I am completely lost... Any help is greatly appreciated. I am unsure where to go to better understand the concepts behind this problem.
The problem: In a class of 50 students, how many students are guaranteed to get the same score on an equally-weighted 20 question quiz? (Assume each question is either completely right or completely wrong - no partial credit)
Since this module pertains to the pigeonhole principle, I assume this question does as well. However, I am unsure how to relate the two in order to solve this problem.
Thank you in advance, wise ones!
First note that there are $21$ possible scores ($\{0, 1, \ldots, 20\}$). Therefore, considering the scores as the pigeonholes and the students as the pigeons (I love the terminology!), we must have, by the pigeonhole principle, $3$ pigeons (students) in one of the holes.
Therefore, three students are guaranteed to get the same score on an equally-weighted twenty question quiz.
If you are not entirely clear on the pigeonhole principle, you could also think about it like this: assume that we could make it such that just $2$ students are guaranteed to get the same score. But that would mean that each score (hole) could contain at most $2$ students, which implies that there are at most $2 \cdot 21 = 42 < 50$ students, which is obviously incorrect. Conversely, we can evidently construct a case where no one score contains four students ($8$ scores containing $3$ students, $13$ scores containing $2$ students).