SIMON-MARAIS 2021 B1

Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$ . Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$ . Call the resulting matrix $A$ . Determine the probability that $A^2 = O$ , as a function of $n$ .