Explain what is meant by the terms boson and fermion.

Three distinguishable spin-1 particles are governed by the Hamiltonian

$$H=\frac{2 \lambda}{\hbar^{2}}\left(\mathbf{S}_{1} \cdot \mathbf{S}_{2}+\mathbf{S}_{2} \cdot \mathbf{S}_{3}+\mathbf{S}_{3} \cdot \mathbf{S}_{1}\right)$$

where $\mathbf{S}_{i}$ is the spin operator of particle $i$ and $\lambda$ is a positive constant. How many spin states are possible altogether? By considering the total spin operator, determine the eigenvalues and corresponding degeneracies of the Hamiltonian.

Now consider the case that all three particles are indistinguishable and all have the same spatial wavefunction. What are the degeneracies of the Hamiltonian in this case?