In inelastic collisions, kinetic energy of the system is not conserved but the momentum is.

Kinetic energy is: $0.5 \times \text{mass} \times \text{velocity}^2$. Momentum is: $\text{mass}\times\text{velocity}$.

So, if mass is assumed to be constant, the velocity of the centre of mass of the system has to be different after the collision for the kinetic energy to be different.

However, if the momentum of the system is conserved, the velocity of the centre of mass of the system should remain the same.

So, how can there be a change in kinetic energy of the system if there is no change in momentum?