(a) In Example 5, a mathematical model is discussed which claims to give a distribution of identical balls into boxes in such a way that all distinguishable arrangements are equally probable (Bose-Einstein statistics). Prove this by showing that the probability of a distribution of N balls into n boxes (according to this model) with N1 balls in the first box, N2 in the second, ⋯, Nn in the nth, is 1/C(n−1+N,N) for any set of numbers Ni such that ∑i=1nNi=N.
(b) Show that the model in (a) leads to Maxwell-Boltzmann statistics if the drawn card is replaced (but no extra card added) and to Fermi-Dirac statistics if the drawn card is not replaced. Hint: Calculate in each case the number of possible arrangements of the balls in the boxes. Do first the problem of 4 particles in 6 boxes as in the example, and then do N particles in n boxes (n>N) to get the results in Problem 19.