Theme
Counting the same thing twice — page 4
One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.
Discrete
The largest family that always meets
Change the question from "no two comparable" to "every two share an element" and the answer changes shape. The best antichain is a whole layer; the best intersecting family is a star, and the proof is a circle.
Discrete
How many ways to sort it
An order says some things come before others and leaves the rest open. Counting the orderings consistent with it measures how much is still unknown — and the counting is as hard as any counting problem gets.