For the purposes of this question, I'll limit the question to quantum electrodynamics, although this question could just as well be asked of electroweak theory and QCD.
For a particular order of perturbation theory, how do I know when I have drawn all the Feynman diagrams? For example, I have heard that in electron-electron scattering, at second order, we have t-channel and u-channel diagrams. But how did we know to draw these two channels - I would have stopped at just the t-channel diagram. And besides, if this is second order, what ever happened to first order diagrams of moller scattering?
One factoid I know about how many diagrams to draw is: in QED, a diagram that is third order in $\alpha$ has three vertices. Each vertex has one in-going and one out-going fermion and one virtual photon. A diagram that is fourth order in $\alpha$ has four vertices and so forth. I suppose I could use simple combinatorics to know when I've exhausted all the vertex combinations, but I don't know how to tell when two graphs are redundant, and there's still the t-channel and u-channel mystery as explained above.