When all the elements of the multiset {1, 1, 2, 2, 3, 3, ... , n, n} are placed randomly in the cells of an m × k rectangular array (where mk = 2n), what is the probability P_m,k(p) that exactly p ∈ [0, n] of the pairs are found with their matching partner directly beside them in a row or column — thus forming a 1×2 domino? For the case p = n, this reduces to the domino tiling enumeration problem solved by Kastelyn and which produces the Fibonacci sequence for the well-known case m = 2. In this paper we obtain a formula for the first moment of the probability distribution for a completely general array, and also give an inclusion-exclusion formula for the number of 0-domino configurations. In the case of a 2 × k rectangular array, we give a bijection between the (k−1)-domino configurations and the nodes of the Fibonacci tree of order k, thus finding that the number of such configurations is equal to the path length of the tree; secondly we give generating functions for the number of (k−l)-domino configurations for l ≤ 2 and conjecture results for l ≤ 5. These generating functions are related to convolutions of Fibonacci numbers.