Counting circuits with constraints
In order to bound the number of functions computed by circuits of size $k$, you have at least two options:

Construct a large number of circuits of size $k$, which by construction compute different functions.
Consider a natural probability distribution on circuits of size $k$, and estimate the probability that two random circuits compute the same function.

As an example, it is known that every function on $m$ variables can be computed by a circuit of size $O(2^m/m)$. By considering functions of the form $f_1(x_1,\ldots,x_m) \lor \cdots \lor f_{n/m}(x_{n-m+1},\ldots,x_n)$, this shows that there are at least $(2^{2^m})^{n/m}$ different functions computed by circuits of size $k = O(n2^m/m^2)$. In terms of $k$, the number of functions is roughly exponential in $k\log k$, for $m \gg \log n$.