I have been given the following question: How many different automata $\newcommand{\perm}[1]{\left\langle #1 \right\rangle}A = \perm{Q, \Sigma, \delta, q_o, F}$ is it possible to form, when the number of states $\newcommand{\abs}[1]{\left| #1 \right|}\abs Q = m$ and the size of the alphabet $\abs\Sigma = n$.
Now this should probably be a simple exercise in combinatorics, but the problem is, I don't know what makes two automata different. What exactly is meant by this?
I can state the following:
The automata all have at least the same states $Q$, alphabet $\Sigma$, initial state $q_0$ and final states $F$.
We have $\abs Q\abs\Sigma$ different $Q$--$\Sigma$ pairs, assuming we can transition out of each state with each of the alphabet.
At the same time, as the state transition function $\delta$ maps from $Q \times\Sigma$ to $Q$, we have $\abs Q^{\abs Q\abs\Sigma}$ different kinds of state transition functions.
Assuming every state in $Q$ can be an initial state, the total number of state functions then has to be multiplied by the number of states, so we have $\abs Q \times \abs Q ^{\abs Q \abs\Sigma}$.
The power set of the states $Q$, $\mathit 2^{Q}$, has $2^{\abs Q}$ elements. This essentially gives us the number of combinations of states $F$ that are reachable from the start state in the state transition chain of each machine.
Taking all of these fact into account, I end up with $\abs Q \times \abs Q^{\abs Q \abs\Sigma} \times 2^{\abs Q}$ "different" machines. But are these machines really different?
A thorough discussion of your problem can be found in the article [1], which also contains an extensive bibliography.
[1] Frédérique Bassino and Cyril Nicaud, Enumeration and Random Generation of Accessible Automata, Theor. Comp. Sci. 381(1-3): 86-104 (2007)