Looking for a very fast/"smart" way to compute this number (it was a question asked on an hour-long exam I recently took, so listing everything out for each element in $S_5$ was not an option since I wanted to do the other questions :P). Is there a key observation I am missing that enables us to find this number without much excessive computation, and if so, what is it?
(Note: this is for an introductory class in Abstract Algebra.)
2 orbits. $A_5$ is a subgroup of index 2. Firstly, $A_5$ is an orbit itself, since for any $g,h\in A_5$, there is some $k\in A_5$ such that $kg=h$. The map $A_5\rightarrow S_5$ given by $g\mapsto g.(12)$ gives the other orbit.