This question came up while tutoring: "*$55$ numbers are randomly selected from the first $100$ positive integers. Show there must be two numbers in the list whose difference is $12$.*"

It definitely requires the Pigeonhole Principle but we were unsure how to set it up. My idea was to list all of the possible pairs whose difference is $12$:

$$(1,13), (2,14), \ldots (87, 99), (88, 100)$$ of which there are $88$ pairs. I'm not sure what to do from here.

I know similar questions have been asked here, like [this one](https://math.stackexchange.com/questions/1554338/70-distinct-positive-integers-that-are-%E2%89%A4-200-there-must-be-two-whose-difference). Is there a way to apply those techniques to this problem?