Number of Substitution boxes How many possible 16 bit S-boxes exist? (AES uses 8 bit substitution boxes)
I first taught of calculating 128(keysize in bits)/8(bits) =16, but I believe it's wrong.
An s-box is a bijective function $f: \{0,1\}^b\rightarrow\{0,1\}^b$. This reduces the question to "how many of those $f$ exist".

To see this easily, imagine domain and image of this function as two boxes with $2^b$ elements. How many different sets of arrows can you imagine between those two sets?

For the first arrow, you have $2^b$ choices. For the second arrow, you have $2^b -1$ choices. For the third, $2^b-2$, and so forth to the last arrow, where there's one choice left.

Multiplying those together, you would get $2^b(2^b-1)(2^b-2)\dots 1=2^b!$ possible s-boxes, so in your 16-bit case that makes $2^{16}!$ possibilities.

To put this more mathematically, an s-box is a permutation of $2^b$ elements, and there are $2^b!$ such permutations.