This is for a physics homework but the problem is essentially combinatorial in nature. There are 5 identical particles. Each can have an energy of 1,2,3 or 4. The energy of a particle is a function of its "state", and it is possible that to two different states correspond the same energy. As a matter of fact, we are told that there is 1 state of energy 1, 3 of energy 2, 4 of energy 3 and 5 of energy 4. If we call state of the system a particular combination of states that the particles are in, how many states of the system are there, provided that the total energy of the system (i.e. the sum of the individual energies of all five particles) is 12? Note that since the particles are identical, we do not make a distinction btw the subset of system states arising from the case "particle 1 has energy 1 and particle 2 has energy 3" and the subset arising when "particle 1 has energy 3 and partcile 2 has energy 1". It seems none of the tools of combinatorics are fit for this problem.