Pigeonhole Principle, find the total number

Question

There are 15 different coffee flavours at the cafe. Oddly, each student in my 8 am class has a favourite flavour there. There are just enough students in the class so you can be absolutely sure that 4 students all have the same favourite. How many students are there in the class?

Answer 1

There are $15$ pigeon holes. That is, $15$ flavors of coffee. In this situation, the pigeons are the students. The intuitive idea here is that we want to prolong having $4$ students in any "hole" for as long as possible.

The way I remember this method is take a piece of paper, and draw $15$ boxes. Each box represents the group of students that like a certain flavor of coffee. Now, start from box $1$ and go through box $15$, putting one tick mark (or student) in each box. Now, we have $15$ students, one for each flavor. Now, let's fill the boxes two more times. We now have $45$ students, $3$ in each box. Now, suppose we have a $46^{th}$ student. This student likes some flavor of the $15$ by assumption, and so s/he belongs in one of the boxes.

So, there absolutely must be one box with $4$ students when we surpass $3*15$ by one. That is, there are at least $$3*15+1=46$$ students in the class.

Answer 2

The generalized pigeonhole principle says

If $N$ objects are placed into $k$ boxes, then there is at least one box containing $\lceil\frac{N}k\rceil$ objects

So in this case the "boxes" are the flavors, and the "objects" are the classmates who favor the flavors. So $$\lceil N/15 \rceil=4$$ Now the $N$ classmates must be at least $46$.

This should make intuitive sense because with $N=45$ you could have that each flavor is favored by $3$ students each, which does not satisfy the condition.