A large wedge rests on a horizontal frictionless surface, as shown.  A block starts from rest and slides down the inclined surface of  the wedge, which is rough.  During the motion of the block, the center of mass of the block and wedge

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$\textbf{(A)}\ \text{does not move}$

$\textbf{(B)}\ \text{moves horizontally with constant speed}$

$\textbf{(C)}\ \text{moves horizontally with increasing speed}$

$\textbf{(D)}\ \text{moves vertically with increasing speed}$

$\textbf{(E)}\ \text{moves both horizontally and vertically}$
1. **Identify the system and forces involved:**
   - The system consists of a block and a wedge.
   - The wedge rests on a horizontal frictionless surface.
   - The block slides down the inclined surface of the wedge, which is rough (indicating friction between the block and the wedge).

2. **Analyze the motion of the block and the wedge:**
   - As the block slides down the inclined plane, it exerts a force on the wedge due to gravity and friction.
   - By Newton's third law, the wedge exerts an equal and opposite force on the block.
   - Since the surface under the wedge is frictionless, the wedge will move horizontally in response to the block's motion.

3. **Determine the center of mass of the system:**
   - The center of mass of the system (block + wedge) must obey the conservation of momentum.
   - Initially, the system is at rest, so the initial momentum is zero.
   - As the block slides down, it gains a component of velocity down the incline and horizontally.
   - The wedge moves horizontally in the opposite direction to conserve the horizontal momentum of the system.

4. **Vertical and horizontal components:**
   - Vertically, the center of mass of the system will lower as the block descends the wedge.
   - Horizontally, the center of mass of the system will shift due to the horizontal motion of both the block and the wedge.

5. **Conclusion:**
   - The center of mass of the block and wedge system will move both horizontally and vertically as the block slides down the wedge.

The final answer is $\boxed{\text{E}}$.