I am taking a probability course and we have been of course learning the classic problems of men with hats or variations of that in some other form. The problems being of the type if n men have n hats and mix them all together, whats the probability 0, 1, 2, or k men choose there hat. But from what I can gather is that they are all problems of the same type, for any given random permutation of k objects how many fixed points are there (we weren't taught this way, just my observations). My professor taught us to solve this using conditional probabilities (conditioning on whether the first person matched and then generating a system of equations) and using the inclusion-exclusion principle. But is there any other way to reason this?
Thanks for reading..