The state vector grows by 2^N so 5 qubits is 2^5 = 32 continuous variables. I am not sure where you are getting a factorial from.

The mathematical description of a quantum computer is pretty much identical to that of a probabilistic computer (a kind of classical computer) with the only real difference being that real probabilities, which range from 0 to 1, are replaced with probability amplitudes, which are complex-valued, so they can be negative or even imaginary and only their square-magnitudes need to range from 0 to 1.

In a probabilistic computer, you describe the system with a vector that contains the real probabilities for every possible outcome, so if you have 3 bits, then the vector needs to be of size 2^3=8 since it needs to hold the probabilities for 000, 001, 010, 011, 100, 101, 110, and 111. You then evolve the vector by multiplying Markov matrices by the vector, which represents how a logic gate affects the probabilities. A Markov matrix is nothing more than a probabilistic truth table.

Again, quantum information evolves exactly the same way except that the vector contains complex-valued probability amplitudes as opposed to real probabilities and that the Markov matrices are replaced by unitary matrices, which the only difference is that unitary matrices, again, contain probability amplitudes rather than real probabilities.