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''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 :)

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

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