FI-MO 1999 Problem 5

An ordinary domino tile can be identified as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$ Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile. We say that two domino tiles match, if they have a common component. Generalized n-domino tiles $m$ and $k$ can get values $0, 1,... , n.$ What is the probability that two randomly chosen $n$ -domino tiles match?