Question: > 
> There is a group of finite aliens on a spaceship. Show that there are at least $2$ aliens who know the same number of aliens on the spaceship. 
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I was given a hint, and that was to use the pigeonhole principle. I think I can visually see it but I am unsure how to show it.

Answer: 
HINT: Let $n$ be the number of aliens on the ship. As noted in the comments, you must assume that $n\ge 2$. Suppose that A is an alien on the ship; how many of the others can it know? The largest possible answer is $n-1$, and the smallest is $0$. 

* How many possibilities is that?
* Is it actually possible for one of the aliens to know $n-1$ of the others and another to know $0$?
An alien can know between $0$ and $n-1$ aliens. That is $n$ possibilities. If one alien knows everyone else, then there is no one who doesn't know any one. And the other way around. So possibilities $0$ and $n-1$ are mutually exclusive. So we are down to $n-1$ possibilities for each alien to know others.

So if there are $n$ aliens and $n-1$ possibilities of knowing others, with then at least $2$ aliens exist who know the same number of aliens on the spaceship.