We are used to collisions where either linear momentum is conserved, such as two billiard balls colliding, or just angular momentum is conserved, such as when a kid jumps and lands on a merry-go-round on a playground. We are not used to seeing cases where BOTH types of momentum are conserved. But this can happen!
When there are no net forces acting on a system, linear momentum is conserved for the system.
When there are no net torques acting on a system, angular momentum is conserved for the system.
This can happen for a rod lying on a sheet of ice or air table, where there is no friction and gravity does not affect the horizontal motion. No part of the rod is nailed down, so the entire rod is free to move, and not just rotate. If a particle comes flying in and sticks to the rod, then we would have both linear motion and rotational motion of the system! And both momenta would be conserved.
A key thing to take away from this example is that there is a new center of mass of the system. This is the key point in the problem since i) this is the point that will move along a straight line after the collision, and ii) it is the axis of rotation of the system after the collision! Check out how to set this bugger up, and see if it makes sense.