DE MOIVRE. 151 



1 



chance of taking a first is ^ ; and tliere are then five things left, 



and the chance of now taking 5 is ^ . Therefore the required 



chance is -^ . Then De Moivre says, 



Since the taking a in the first i)lace, and h in the second, is hut one 

 single Case of those by which six Things may change their order, being 

 taken two and two ; it follows that the number of Changes or Permu- 

 tations of six Things, taken two and two, must be 30. 



275. In his Preface De Moivre says, 



Having explained the common Rules of Combinations, and given a 

 Theorem which may be of use for the Solution of some Problems re- 

 lating to that Subject, I lay down a new Theorem, which is properly a 

 contraction of the former, whereby several Questions of Chance are 

 resolved with wonderful ease, tho' the Solution might seem at first sight 

 to be of insuperable difiiculty. 



The mil} Theorem amounts to nothing more than the simplifi- 

 cation of an expression by cancelling factors, which occur in its 

 numerator and denominator ; see Doctrine of Chances, pages ix. 89. 



27C. Problems xxi. to XXV. consist of easy applications to 

 questions concerning Lotteries of the principles established in the 

 Problems xv. to XX. ; only the first two of these questions con- 

 cerning Lotteries appeared in the first edition. 



A Scholium is given on page 95 of the third edition which 

 deserves notice. De Moivre quotes the following formula : Sup- 

 pose a and n to be positive integers ; then 



1111 1 



- + — r-T + — -^ + — 7-i. + ...+ 



n n-\-l 71 + 2 ?i + 3 a — 1 



_ a 1 1 A (l^ 1\ B (1 r 



~ ^""^7^ + 2;^ "" 2^ "^ 2 \n' " ^; "^ I W a\ 



where A=\. ^^'W ^=A' - 



