The answer is (c): the balance board will remain balanced, as seen in an mpeg video by clicking your mouse on the photograph below.
Because the mass of the cart on the right is twice that of the cart on the left, the cart on the right will leave the "collision" with half the velocity of the cart on the left. Therefore the beam will remain in balance until the lighter cart hits the end, after which time the board will tilt down to the right.
