Question 658660
Hint:


Parts a) and b) are mutually exclusive and complements of each other. In other words, 


P(At least one gets their own cap) + P(No one gets their own cap back) = 1


So if you can find one of the probabilities above, then you can automatically find the other.


It's easiest to find P(No one gets their own cap back) and you can use a <a href="http://en.wikipedia.org/wiki/Derangement">derangement</a> to do so.


A derangement is a permutation in which something like ABCDE rearranges to EDABC. This new arrangement is where all letters have moved (no single letter is in the same spot).