【Example 7】 How many different subsets does an $n$-order set $A$ have? How many of these are proper subsets?
For each subset of $A$, some elements of $A$ may belong to this subset, while others may not. This is akin to making a choice between "belonging" and "not belonging" for each element in $A$, so it is a problem of repeated permutations, resulting in a total of $2^{n}$ different subsets.

When every element of $A$ belongs to a certain subset, this subset is $A$ itself, and $A$ is not a proper subset of itself, so a set of order $n$ has a total of $2^{n}-1$ different proper subsets.