How many size $s$ circuits from $\{0, 1\}^n \to \{0, 1\}$ are there?
Hint: Each of the $s$ gates is connected to two other gates or inputs, for a total of $(s+n)^2$ possibilities. This gives a rough upper bound which isn't tight, but is good enough for many purposes, for example to show that some functions require circuits of size $\Omega(2^n/n)$. See Steurer's notes, for example.