KöMaL 2015 B4722

Each permutation of an $n$ -element set is coloured in either red, white or green. Let $N_{\rm RWG}$ denote the number of ways to perform a red permutation followed by a white permutation and then a green permutation, such that each element is restored to its initial position at the end. Analogously, let $N_{\rm GWR}$ denote the number of ways to perform a green permutation followed by a white permutation and then finally a red permutation, such that each element is restored to its initial position. Show that $N_{\rm RWG}=N_{\rm GWR}$.