# In need of interpretation of a ordered-subset related problem.

''Given a set $$S_n = \{1, 2, 3, \ldots, n\}$$, we define a preference list to be an ordered subset of $$S_n$$. Let $$P_n$$ be the number of preference lists of $$S_n$$. Show that for positive integers $$n > m$$, $$P_n - P_m$$ is divisible by $$n - m$$.

Note: the empty set and $$S_n$$ are subsets of $$S_n$$.''

Can anyone help me out to make me understand the problem ? (giving an explained example of this question will be very good ) Thanks in advance :)

For example, for $$n=3$$ the preference lists of $$\{1,2,3\}$$ are $$\emptyset\\ (1)\\ (2)\\ (3)\\ (1, 2)\\ (2, 1)\\ (1, 3)\\ (3, 1)\\ (2, 3)\\ (3, 2)\\ (1, 2, 3)\\ (1, 3, 2)\\ (2, 1, 3)\\ (2, 3, 1)\\ (3, 1, 2)\\ (3, 2, 1)$$