See attached file.
You can solve this problem as follows. For each of the 16 particles you introduce a variable that indicates the energy divided by epsilon of its quantum state it is in. Let's call this variable E_i for particle nr. i. Then the E_i are integers. The total energy of the system of 16 particles is 18 epsilon, therefore:
E_1 + E_2 + E_3 +....+ E_16 = E (1)
where E = 18
Any solution of equation (1) with E_i integers larger than or equal to zero defines a possible microstate of the system with energy E. Also two different solutions define different microstates (assuming that the particles are not identical) So, the number of microstates is the number of solutions of equation (1) for E = 18. To count the number of solutions, let's consider a different problem described by the same equation. Suppose during ...
A detailed solution is given.