A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system ?

The toddler is randomly going along a trolley that is moving at velocity V.  However, the child's running will have no impact on the trolley's centre of mass's velocity because the boy's action only creates internal force. Any internal force will not result in any change in the velocity of the trolley's centre of mass, only external forces can alter a system's centre of mass.

Hence the correct answer is remains unchanged.