Given a finite set X of elements, divided into disjoint subsets, we define a derangement of X as a permutation which leaves none of the elements in their original subsets. The probability of a random permutation being a derangement is discussed, particularly its asymptotic value as the cardinality of X and the number of subsets tend, under certain conditions, to infinity. Finally, the problem is extended to studying the number of elements which are transferred by a general permutation to a subset other than their initial one.