Six children ($a$ through $f$) are playing on a carousel with 6 seats such that $a$ is sitting in front of $d$, $b$ is sitting in front of $e$, and $c$ is sitting in front of $f$.
How many ways are there to change the sitting order, such that no child is sitting in front of the child they are sitting in front of now?
So for example, if child $a$ is sitting in seat number $1$, child $b$ is sitting in $2$ and so on, how many combinations are there such that $a$ is not in front of $d$, $b$ is not in front of $e$ and $c$ is not in front of $f$.
It is a case of counting possibilities. $a$ can go in $6$ seats. $d$ then has four choices, but you need to distinguish whether the remaining seats are in a group of $4$, groups of $1,3$ or groups of $2,2$. For each one, see how many ways there are to position $b,e$. Have you tried?