Permutations and Combination Locks

Consider a combination lock with n buttons, numbered 1 through n. A valid combination consists of a sequence of button-pushes, in which each button is pushed exactly once. If the buttons must be pushed one at a time, then clearly there will be n! possible combinations. But what if we are allowed to push butt6ns simultaneously? We can represent a valid combination for such a lock as a sequence of disjoint, nonempty subsets of the set B = {1,2,. . ., n} whose union is B. Each set in the sequence specifies a collection of buttons to be pushed simultaneously. For example, if n = 3 then we have the following possible combinations: