[14] Define the occupancy numbers for n balls distributed over k cells as a k-tuple of integers (n1, n2,...,nk) satisfying n1+n2+···+nk =

n with ni ≥ 0 (1 ≤ i ≤ k). That is, the first cell contains n1 balls, the second cell n2 balls, and so on.

(a) Show that there are n n1,...,nk placements of n balls in k cells resulting in the numbers (n1,...,nk).

(b) There are kn possible placements of n balls in k cells altogether.

Compute the fraction that results in the given occupancy numbers.

(c) Assume that all kn possible placements of n balls in k cells are equally probable. Conclude that the probability of obtaining the given occupancy numbers is n!

n1! ··· nk!

k−n.

Comments. In physics this is known as the Maxwell–Boltzmann statistics

(here ‘statistics’ is used as a synonym for ‘distribution’). Source: [W.

Feller, Ibid.].