A derangement is a permutation that has no fixed points. In this paper, we study a subclass of derangements on \(r+ n\) objects such that none of the first r objects returns to the first r objects location. We determine explicit formulas and recurrence relations for the number of this subclass of derangement. Also, the probability that a permutation is in this subclass of derangements is close to 1/e. Finally, we will present the Stirling transformation formula of these numbers.
