ILL 1985 Problem 80

Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$ . If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$ , compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$ . Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$