Orbit counting
Theorem. Let $G\leq \text{Sym}(\Omega)$. Then the number of orbits of $G$ on $\Omega$ is
\[\frac{1}{|G|}\sum_{g\in G}\text{fix}(g).\]Proof. Form a bipartite graph, with the vertex set $G\cup\Omega$.
Theorem. Let $G\leq \text{Sym}(\Omega)$. Then the number of orbits of $G$ on $\Omega$ is
\[\frac{1}{|G|}\sum_{g\in G}\text{fix}(g).\]Proof. Form a bipartite graph, with the vertex set $G\cup\Omega$.