TITLE: Looking for a problem where one could use a cardinality argument to find a solution.
QUESTION [12 upvotes]: I would like to find an exercise of the type: Find some $x$ in $A\setminus B$. Solution: since $A$ is uncountable and $B$ is countable such $x$ exists...

REPLY [4 votes]: One example which I quite like is this:

Show that there are points in the plane which cannot be constructed from unit segment using just compass and straightedge.

The cardinality based proof is simple: We are starting with two points, at each step of construction we can only add finitely many new points and we employ constructions which only have finitely many steps. So the set of all constructible points is countable. The set of all points in the plane has cardinality of continuum, it is uncountable.
The students will probably see in some algebra course1 proofs showing that some particular lengths (such as $\sqrt[3]2$ or $\cos\pi/9$ cannot be constructed). But this proof can be considered simple and it also works for other constructions of similar type. (I.e. constructions with finitely many steps where only finitely many new objects are added in each step.)
1 I would guess that typically cardinalities appear in a curriculum before the course of algebra which includes result needed for proving impossibility of trisecting an angle or doubling a cube. But perhaps somewhere these topics are taught in a different order.