Combinatorics Generalized Pigeonhole Principle question

Question

I cant solve this exercise / I dont understand it:
There are n kids and 3 possible classes.
Every couple of kids are going to the same class,
prove that there is a class that 2/3 of the kids are going to.

Answer 1

Let $n$ be the number of kids, and let $x\ge y\ge z$ be the class sizes. We have to show that $x\ge\frac23n$.

We may assume that each kid is in at least two classes, for if some kid were in only one class, then they weould all have to be in that class.

Let $p$ be the number of pairs $(C,K)$ consisting of a class and a kid in that class.

Counting then one way, we clearly have $$p=x+y+z\le3x.\tag1$$

Counting them another way, since there are $n$ kids and each kid is in at least two classes, $$p\ge2n.\tag2$$

Combining the inequalities $(1)$ and $(2)$, we get $$3x\ge2n,$$ that is, $$x\ge\frac23n.$$