Solution: 2019 Winter Midterm - 6

Author: Michiel Smid

Question

Let $n \geq 1$ be an integer. A group of $n$ students write an exam. Each student receives a grade, which is an element of the set $\{A, B, C, D, F\}$.
What is the minimum value for $n$, such that there must be at least four students who receive the same grade?

(a)

(b)

(c)

(d)

Solution

We can use the pigeonhole principle to solve this problem.

We need to have 1 more than the maximum number of possible grades.

That way, we can guarantee that if all students receive different grades, the last student will receive a duplicate grade.

There are 5 possible grades: $ A, B, C, D, F $

Thus, the minimum value for $ n $ is $ 5 \cdot 3 + 1 $

Thus, the minimum value for $ n $ is 16.