ICO DE MOIVKE. 



{p + 1)" 



(_p+l)"_/^« + /01^(^_l)» 



_ /(/-lK/-2) (^_,)„^.. 



Be Moivre infers this general result from the examination 

 of the simple cases in which f is equal to 1, % 8, 4 respec- 

 tively. 



He says in his Preface respecting this problem, 



When I began for the first time to attempt its Solution, I had 

 nothing else to guide me but the common Kules of Combinations, such 

 as they had been delivered by Dr. Wallis and others; which when I 

 endeavoured to apply, I was surprized to find that my Calculation 

 swelled by degrees to an intolerable Bulk : For this reason I was forced 

 to turn my Views another way, and to try whether the Solution I 

 was seeking for might not be deduced from some easier considerations; 

 whereupon I happily fell upon the Method I have been mentioning, 

 which as it led me to a very great Simplicity in the Solution, so I 

 look upon it to be an Improvement made to the Method of Com- 

 binations. 



The problem has attracted much attention; we shall find it 

 discussed by the following writers : Mallet, Acta Helvetica, 1772 ; 

 Euler, Opuscula Analytica, Vol. ii. 1785; Laplace, Memoir es.., 

 2Mr clivers Savans', 1774, Theorie... cles Proh. page 191 ; Trembley, 

 Memoires de V Acad... Berlin, 1794, 1795. 



We shall recur to the problem when we are giving an account 

 of Euler's writings on our subject. 



292. Problem XL. is as follows : 



To find in how many Trials it will be probable that A with a Die 

 of any given number of Faces shall throw any proposed number of 

 them. 



1 



Take the formula given in Art. 291, suppose it equal to ^ , 



and seek for the value of n. There is no method for solving 

 this equation exactly, so De Moivre adopts an approximation. 

 He supposes that ^ + 1, ^, ^ — 1, j9 — 2, are in Geometrical 



