Example (something I raised my hand to ask when I was in secondary school):
How do you know that $\dfrac{a!}{b!(a-b)!}$ is necessarily an integer whenever $a$ and $b$ are natural numbers such that $a>b$? This is a problem in elementary number theory.
Though there are other ways to prove it, the best makes use of an observation in combinatorics: given any set $T\subset\{1,2,\ldots,a\}$ of cardinality $b$, the number of permutations of $\{1,2,\ldots,a\}$ such that the first $b$ terms of the sequence form the set $T$ is exactly $b!(a-b)!$.
Combinatorics $\to$ elementary number theory
Could you give more examples, suitable for e.g. first-year undergrads or an exceptionally mathematical high school, where you prove a statement in one area of maths by going to a totally different area of maths?
Nothing too advanced: e.g. using complex contours to evaluate a real integral, or using topology to prove properties of the free group.