Function that cannot be computed by a Boolean circuit of size $2^n/2n$

Show that, for sufficiently large $n$, there is a function $f\colon\{0,1\}^n \to \{0,1\} $ that cannot be computed by a Boolean circuit with fan-in $2$ with $\frac{2^n}{2n}$ gates. Please give me a hint.

We need counting argument here. First thing is, there are $2^{2^{n}}$ many boolean functions, second thing is to count the number of boolean circuits for $k$ size boolean circuit. There are $2^{k^2} \times 3^{k}$ many boolean circuits of size $k$ (use adjacency matrix ). In your case value of $k =\frac{ 2^n}{2n}$, now you can easily check that


$$2^{2^{n}} > 2^{k^2} \times 3^{k} $$

So it means there are more number of functions than the total number of boolean circuits. There has to be at least one function that can not be computed by $k$ size circuit ( pigeonhole principle ). One thing to note here that I have proved the existence of such type of boolean circuit but to come up with such type of boolean circuit is quite hard.

StackExchange Computer Science Q#72483, answer score: 4