Find a one-to-one correspondence between the permutations of the set $\{1, 2,\dots,n\}$ and the towers $A_0 \subsetneq A_1 \subsetneq A_2 \subsetneq\dots \subsetneq A_n$, where $\vert A_k\vert = k$ for $k = 0, 1, 2,\dots,n$.
The permutations of the set $\{1, 2,\dots,n\}$ should be $n!$:
There are $n$ ways to assign $1$st element, There are $n-1$ ways to assign $2$nd element, .... There is $1$ way to assign $n$th element
$\Rightarrow n!$ permutations
But the number of proper subsets of a set should be $2^n-1$ which is not equal to $n!$