XXIII.—On a Problem in Combinations

Extract

Several years ago, when discussing the question of the distribution of the stars, a problem occurred to Professor Forbes, which, simple as it is, appears to have escaped notice prior to that time. Having been consulted as to its solution, I communicated my results to Professor Forbes, who has inserted one of them in his paper printed in the Philosophical Magazine for 1850, vol. xxxvii., p. 425. But for the very ingenious application which Professor Forbes has there made of it, the problem might probably not be worth recurring to. As it is, I have thought it would not be altogether uninteresting to give the complete solution.

The Problem is as follows:—There are n dice, each of which has p faces, p being not less than n; it is required to find the number of arrangements which can be formed with them, 1°, So that no two show the same face; 2°, That no three show the same face; 3°, That no four do so, and so on.