Mathematics
Generating Functions for Solving Combinatorial Counting Problems
Quick fact
The generating function for the sequence 1, 2, 3, 4, ... is 1/(1-x)^2, and extracting the coefficient of x^n immediately gives n+1 — a result that would otherwise require summing a series of natural numbers.
Why this is interesting
You've probably seen how polynomials like (1+x)^n can be expanded, but did you know they hide a powerful secret for counting? By turning a counting problem into a polynomial or infinite series, you can use algebra to solve it, even when the answer would otherwise be a tangled mess of recurrences.