Answer
Path ranking is $4\gt6\gt7\gt1\gt2=3=5$.
Work Step by Step
Because the mass and speed of every gum are the same. The factor that differentiates every path is the distance of the throwing force acting perpendicular to the center of the rotation.
The longer the distance from the point $O$ in a perpendicular way, the greatest the angular momentum the slam and the gum would have.
So, the ranking is $4\gt6\gt7\gt1\gt2=3=5$, with the path $2,3,5$ resulting in $0$ angular momentum because this path leads directly to the center of rotation $O$.
