The result in 
is very important, it is
essential in the proof of
the following counting lemma which, together with later refinements,
forms the basic tool of the theory of enumeration under finite group
action:
.
The Lemma of Cauchy-Frobenius
The number of orbits of a finite group 
acting on a finite set 
is equal to the average
number of fixed points:
Proof:
which is, by , equal to
.
Now you can try to make some calculations using the Cauchy-Frobenius Lemma.