Denote by $S$ be the set of all binary strings of length $n$. Of course $|S|=2^n$. Let $S'\subset S$ be the set of our formatted expressions of length $n$. For any $s'\in S'$, denote by $W(s')$ the set of all binary strings of length $n$ which differ from $s'$ in at most one bit. $W(s')$ consists of $s'$ itself plus another $n$ strings, each one differs from $s'$ only in the $i$-th bit, $i=1,2,\dots,n$. Hence $|W(s')|=n+1$. Let $s_1',s_2'\in S', s_1'\neq s_2'$. Then $W(s_1')\cap W(s_2')=\emptyset$. Indeed, if $s_0\in W(s_1')\cap W(s_2')$, it would mean $s'_1$ and $s'_2$ differ in at most two bits, so they cannot be both in $S'$, a contradiction. Thus, $W(s'),s'\in S'$ are disjoint sets, each one of $n+1$ elements. Hence $$|S'|\leq \left\lfloor \frac{|S|}{n+1}\right \rfloor =\left \lfloor\frac{2^n}{n+1}\right \rfloor$$