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. Solution: First, note that this factors into $(y-3)^n(x+7)^n$. Now, each term of $(y-3)^n$ will be distinct from every term of $(x+y)^n$ but critically, a term multiplied by the first $n$th power by a term in the second $n$th power will produce a distinct term in the final sum. We also know the number of terms in each $n$th power will be the same. As $n$ varies, there will be $n+1$ terms in the resulting sum in each of $(y-3)^n$ and $(x+7)^n$ meaning there are $(n+1)^2$ terms in the result of $(y-3)^n(x+7)^n$. We want the lest $n$ such that $(n+1)^2> 1996$ which happens when $n=44$.