Chapter 9, Problem 50E

You have six shelves, one above the other and all above the floor, and six volumes of an encyclopedia, A, B, C, D, E, and F.
(a) List all the ways you can arrange the volumes with five on the floor and one on the sixth/top shelf.
One way might be $\{\text{ABCDE},-,-,-,-,-,\text{F}\}$ .
(b) List all the ways you can arrange them with four on the floor and two on the third shelf.
(c) Show that there are many more ways, relative to parts (a) and (b), to arrange the six volumes with two on the floor and two each on the first and second shelves. (There are several ways to answer this, but even listing them all won’t take forever it’s fewer than 100.)
(d) Suddenly, a fantastic change! All six volumes are volume X−it’s impossible to tell them apart. For each of the three distributions described in parts (a), (b), and (c), how many different (distinguishable) ways are there now?
(e) If the energy you expend to lift a volume from the floor is proportional to a shelf’s height, how do the total energies of distributions (a), (b), and (c) compare?
(f) Use these ideas to argue that the relative probabilities of occupying the lowest energy states should be higher for bosons than for classically distinguishable particles.
(g) Combine these ideas with a famous principle to argue that the relative probabilities of occupying the lowest states should be lower for fermions than for classically distinguishable particles.