Let \(d(n, k)\) denote the number of derangements (permutations without fixed points) with \(k\) cycles of the set \([n] = \{1, 2, \ldots, n\}\). In this paper, a new explicit expression for \(d(n, k)\) is presented by graph theoretic method, and a concise regular binary tree representation for \(d(n, k)\) is provided.