8^ MONTMOET. 



are to be aces, or twos, or threes, ,.., and similarly without specify- 

 ing for the h faces, or the c faces, . . . 



He had given the result for this problem in his first edition, 

 page 137, where the factors B, C, JD, E, F,... must however be 

 omitted from his denominator ; he suppressed the demonstration 

 in his first edition because he said it would be long and abstruse, 

 and only intelligible to such persons as were capable of discovering 

 it for themselves. 



148. On his page 46 Montmort gives the following problem, 

 which is new in the second edition : There are n dice each having 

 /faces, marked with the numbers from 1 to/; they are thrown at 

 random : determine the number of ways in which the sum of the 

 numbers exhibited by the dice will be equal to a given number p. 



"We should now solve the problem by finding the coefficient 

 of x^ in the expansion of 



(a; + 03^ + 03'+ ...+x^Y, 



/I — x^y^ 



that is the coefficient of x^'"' in the expansion of I = J , that is 



in the expansion of (1 — x)'"" (1 — x^y. Let p — n = s; then the 

 required number is 



n (ii+l) ... (n-h s —1) 71 (72 + 1) ... (n+s —f— 1) 



«-/ 



n(n-l) n(n + V) ... (n+ s —2f- 1) 

 1.2 l.s-2/ 



The series is to be continued so long as all the factors which 

 occur are positive. Montmort demonstrates the formula, but in a 

 much more laborious way than the above. 



149. The preceding formula is one of the standard results of 

 the subject, and we must now trace its history. The formula was 

 first published by De Moivre without demonstration in the Be 

 Mensura Sortis. Montmort says, on his page 364, that it was derived 

 from page 141 of his first edition; but this assertion is quite un- 

 founded, for all that we have in Montmort's first edition, at the 

 place cited, is a table of the various throws which can be made 

 with any number of dice up to nine in number. Montmort how- 



