340 MALLET. 



suspected that there was a design to cheat the public, which 

 actually happened. 



634. Mallet makes no reference to any preceding writer on 

 the subject ; but solves the problem in a most laborious manner. 

 He finds the chances that the number of persons without prizes 

 should be 1, or 2, or 3, . . . up to n ; then he knows the advantage 

 of the banker corresponding to each case by multiplying the 

 chance by the gain in that case ; and by summing the results he 

 obtains the total advantage. 



635. One part of Mallet's process amounts to investigating 

 the following problem. Suppose a die with r faces ; let it be 

 thrown s times in succession : required the chance that all the 

 faces have appeared. The number of ways in which the desired 

 event can happen is 



and the chance is obtained by dividing this number by r'. 



This is De Moivre's Problem xxxix ; it was afterwards dis- 

 cussed by Laplace and Euler ; see Art. 448. 



Mallet would have saved himself and his readers great labour 

 if he had borrowed De Moivre's formula and demonstration. But 

 he proceeds in a different way, which amounts to what we should 

 now state thus : the number of ways in which the desired event 

 can happen is the product of [r by the sum of all the homogeneous 

 products of the degree s — r which can be formed of the numbers 

 1, 2, 3, ... 7\ He does not demonstrate the truth of this statement ; 

 he merely examines one very easy case, and says without offering 

 any evidence that the other cases will be obtained by following the 

 same method. See his page 144. 



Mallet after giving the result in the manner we have just indi- 

 cated proceeds to transform it ; and thus he arrives at the same 

 formula as we have quoted from De Moivre. Mallet does not 

 demonstrate the truth of his transformation generally; he contents 

 himself with taking some simple cases. 



636. The transformation to which we have just alluded, 



