Rotational Mechanics Conceptual Question?

A rod of m M and length L rests on a frictionless horizontal surface. A particle also of m M moving with a velocity of V strikes the rod at a distance of L/4 from the center and comes to rest. Then by the conservation of linear momentum, the velocity of the center of m of rod becomes V. And by the conservation of angular momentum ( M * V * L/4 = I * W) we can determine W where W is the angular velocity gained by rod and I is the moment of inertia about the center of m. Thus the total K.E of rod is 1/2 * M * V^2 + 1/2 * I * W^2. Suppose the particle struck exactly at the center of the rod and comes to rest, then too by conservation of linear momentum, the velocity of the center of m of the rod is V but it has no angular velocity. Then the total K.E of rod = 1/2 * M * V^2. We can see that the K.E in the first case is more. But that should not be since energy is conserved in this case. Can anybody explain where am I mistaken in my essment ?