Please help with the stat-mech question below:

In this problem set we will explore some of our different quantum statistics and systems. W'e start with the different possible states and number of states available in our different system types. A set of N = 5 particles of mass m are in the same ID simple harmonic oscillator potential of frequency to. Consider the lowest-energy N = 5particle microstate(s) for this system in the following cases: a The particles are all distinguishable. o The particles are identical spin-U bosons. o The particles are identical spin1 / 2 fermions. (Don't forget to account for the spin!) 0 Extra Part (Not for Credit) The particles are identical spin-1 bosons. (Don't forget to account for the spin!) (a) For each of the cases, determine the energy of the lowest-energy N = 5-particle microstate, the degeneracy / multiplicity of that state, and describe the state by its occupation numbers (eg. how many particles are in the n = 0 state, how many are in the n = 1 state, etc.) Hint {highlight to reveal): [ l When we have indistinguishable particles we only care about the permutation classes. For example, in lecture we looked at the permutatiori classes in the cases when we had N = 2 particles in an A = 2-level system and then N = 3 particles in an A = 3-level system. Next let's look at a system of N = 3 identical particles with A = 4 available states {|1), |2), |3), |4)}. We can list these states either using the balls and bins notation I used in class or by giving the four occupation numbers {n.1, n2, n3. n4} for each of the four states. (b) How many permutation classes do we have in this system? How many bosonic states are in this system? How many fermionic states? Describe each of the possible fermionic states (using either the balls and bins notation or the occupation numbers). Note: We can assume that our four states include spin - e.g. if we are looking at spin-1/2 fermions these four states could be two different energy levels with the two different spin values. If we are looking at spin-0 bosons these four states could just be four different energy levels or two different energy levels with some other source of a factor of two degeneracy. In the high-temperature limit we don't really care about the energies - the important point is that for a single particle a total of four states are available.] In the high-energy limit, each available state has equal probability of occurring. But different states are available based on whether our particles are bosons or fermions. (c) Consider both fermions and bosons in this N = 3 particles in an A = 4-level system in the high-energy limit and determine the probability of finding exactly one particle in the one-particle ground state. That is for both fermions and bosons determine the probability that n1 = 1