Collisions in rotational dynamics

If I have say a stationary rod, and a particle collides with the rod whilst moving at speed v at the top end, why does the rod then proceed to rotate about the centre of mass of the system (whether that be in an elastic or completely inelastic collision where they both stick together).

I believe it may have something to do with the fact the angular momentum can be written as
with the first term that of a particle with the total mass of the system moving at the CM, and the second the angular momentum of the system relative to the CM (i.e in the CM frame). Thanks.

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