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

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The discussion focuses on finding a microcanonical solution for a three-level system with energy states (0, ε, ε) and the challenge of expressing variables A, B, and C solely in terms of total energy E, energy levels ε, and particle number N. The equation for the number of microstates W is presented, but the user struggles to rearrange it effectively, ultimately deciding to ignore terms related to B and C when differentiating W(E). To aid understanding, examples with specific values for N and E are suggested, such as N = 3 and E = 2ε, to explore possible microstates. The conversation emphasizes the complexity of the problem and seeks insights on the approach taken. Overall, the thread highlights the difficulties in applying traditional methods to a three-level system.
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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
 
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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.
 
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