Microcanonical solution for three level system with (0, ε, ε)?

[decahal]

Homework Statement

Some particle has three states A,B,C. A has 0 energy while B and C both have same ε energy. Some system has N such particles. Find the number of possible states W(E) and express the entropy as a function of the total energy.

Relevant Equations

S=klogW

W = N!/(A!B!C!)

S(E)/k = NlogN - AlogA - BlogB- ClogC
I spent a lot of time but find it impossible to arrange the equation to express all A, B, C in terms of only E and ε and N, which is usually how you solve two level systems.
I just gave up and said B and C is completely unrelated to the total energy E so when I partial diff the W(E) I just ignore the last two terms.
Anyone with any insights whether that's the right way to do it?
Thanks

 

[TSny]

To get a feel for the problem, it can help to consider a few simple cases.

Suppose the number of particles is N = 3 and let the energy of the system be E = 2ε.
One possible microstate is (A, B, B) which means particle 1 is in state A, particle 2 is in state B, and particle 3 is also in state B. List all the other possible microstates for this energy E. What is W(E) for this example?

Try N = 3 and E = ε. Then you could try N = 4 and E = 2ε. This might help to see how to think about the number of microstates for arbitrary N and for arbitrary E = mε where m is an integer ≤ N.