#combinatorics

A finite set of points in the plane does not lie on a single line. Prove that some line passes through exactly two of the points.

$n$ guests check their hats, and the hats are handed back in a uniformly random order. Let $p_n$ be the probability that no guest gets their own hat. Find a formula for $p_n$ and its limit as $n\to \infty$.

Fifteen schoolgirls go for a walk every day for a week, in five rows of three. Arrange the rows for the seven days so that no two girls walk in the same row more than once.