1996 AIME Problems/Problem 3

Problem

Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$ , after like terms have been collected, has at least 1996 terms.

Using Simon's Favorite Factoring Trick, rewrite as $[(x+7)(y-3)]^n = (x+7)^n(y-3)^n$ . Both binomial expansions will contain $n+1$ non-like terms; their product will contain $(n+1)^2$ terms, as each term will have an unique power of $x$ or $y$ and so none of the terms will need to be collected. Hence $(n+1)^2 > 1996$ , the smallest square after $1996$ is $2025 = 45^2$ , so our answer is $45 - 1 = 044$ .