# Counting circuits with constraints

Please forgive me if this question is trivial, I couldn’t come up with an answer (nor finding one).

In order to show that there are boolean functions $$f : \{0,1\}^n \rightarrow \{0,1\}$$ which can be computed only using circuits of size $$\Omega(2^n/n)$$, we use a counting argument: there are at most $$O(2^{k \log k})$$ circuits of size $$k$$, and $$2^{2^n}$$ such functions.

Suppose that I am interested in counting circuits of size $$k$$ that compute different functions. The "simple" counting argument won’t work since it may be possible that two "syntactically" different circuits actually compute the same function. In other words, I want to bound the size of the set: $$F = \{ f: \{0,1\}^n \rightarrow \{0,1\} | f \text{ can be computed using a circuit of size }k \}$$

Then $$|F| <$$ the number of circuits of size $$k$$ (since any circuit computes one function), but how can I bound $$|F|$$ from below? (i.e. $$x <|F|$$)