(a) Consider a system that may be unoccupied with energy zero or occupied with energy zero or occupied by one particle in either of two states, one of energy zero and one of energy ε. Show that the Gibbs sum for this system is

Z = 1 + λ + λ exp (- ε/τ)

Our assumption excludes the possibility of one particle in each state at the same time. Notice that we include in the sum a term for N = 0 as a particular state of a system of a variable number of particles
(b) Show that the thermal average occupancy of the system is

= λ + λexp(-ε/τ) / z

(c) Show that the thermal average occupancy of the state at energy ε is

= λexp(-ε/τ) / z

(d) Find an expression for the thermal average energy of the system.
(e) Allow the possibility that the orbit at 0 and at ε may be occupied each by one particle at the same time; show that

z = 1 + λ + λ exp (–ε/τ) + λ2 exp (–ε/τ) = (1 + λ)[1 + λ exp(–ε/τ)]

Because z can be factored as shown, we have in effect two independent systems.