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An alternative approach to a problem by A. de Moivre
Published online by Cambridge University Press: 23 January 2015
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Suppose n (fair) dice each having m faces marked with numbers 1 to m, are thrown at random. The problem of determining the number of ways in which the sum of the numbers exhibited by the dice will be equal to a given number k has a very long history. In particular, the three dice problem (i.e. the case where m = 6, n = 3), goes back to the 13th century (see [1]). Later, it was Cardano and Galileo who solved it (see [2, 3]). The general case was stated without proof in A. de Moivre's first work on probability, De Mensura Sortis, (1712, p.220). De Moivre was the first who published a proof (using generating functions) in Miscellanea Analytica (1730, p. 196). Furthermore, Montmort had also the solution (via inclusion-exclusion) by 1713, and independently of De Moivre (see [4, 2]).
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