o 



12 MALLET. 



This is the general theorem which Mallet enunciates; but only 

 demonstrates in a few simple cases. 



If we put 1, 2, 3, ... r respectively for a, h, c, .,,k we obtain 

 the theorem by which we pass from the formula of Mallet to that 

 of De Moivre, namely, the sum of the homogeneous products of 

 the degree s — r which can be formed of the numbers 1, 2, ... r is 

 equal to 



ij._,,_.,,rfc3),..,_r(t^«fcS,_a,.,...), 



The particular case in which s — r+l gives us the following 



result, 



l+2 + S-\- ...+r 



, ^(r-l) r(r^l)(r- 2) ^^, ] 



+ 1.2 ^"^ ^^ 1.2.3 ^"^ ^^ ^"V 



which is a known result. 



687. When Mallet has finished his laborious investigation he 

 says, very justly, il y a apparence que celui qui fit cette Lotterie ne 

 setoit pas donne la peine defaire tons les calculs pr^cedens. 



638. Mallet's result coincides with that which Montmort gave, 

 and this result being so simple suggested that there might be an 

 easier method of arriving at it. Accordingly Mallet gives another 

 solution, in which like Montmort he investigates directly not the 

 advantage of the director of the lottery, but the expectation of each 

 ticket-holder. But even this solution is more laborious than Mont- 

 mort's, because Mallet takes separately the case in which a ticket- 

 holder has 1, or 2, or 3, . . . , or ?^ prizes ; while in Montmort's 

 solution there is no necessity for this. 



639. Mallet gives the result of the following problem : Re- 

 quired the chance that in p throws with a die of n faces a specified 

 face shall appear just m times. The chance is 



\m p —m n^ 



