2015
Problem - 72
Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.
There are totally $C_{n^2}^2$ possible ways to select two squares. By the conclusion of # 4278, there are $2n(n-1)$ ways to select two adjacent squares. Hence, we have $$\frac{2n(n-1)}{C_{n^2}^2} < \frac{1}{2015}\implies N \ge \boxed{90}$$