Chapter 15.4, Problem 20P

(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 $N_1$ balls in the first box, $N_2$ in the second, $\cdots ,N_n$ in the nth, is $1/C(n-1+N,N)$ for any set of numbers $N_i$ such that ${\displaystyle \underset{i=1}{\overset{n}{\sum }}{N}_i}=N$.

(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. First do 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.