150 DE MOIVRE. 



no order at all; that is if the die were to be thrown and the 

 stake awarded to A, B, or C, according as the face which appeared 

 was one of the a, h, c respectively. If there is to be an order, 

 but the order is as Hkely to be one as another, the result will be 

 different. The chance of A for example will be one sixth of the 

 sum arising from six possible and equally likely cases. It will be 

 found that A's chance is 



a 



{6a' + 9a(h-i-c) + S (h' + c') + She} 

 Q{n'-{b + c) (c + a) (a + h)} 



272. Problem xii. appeared for the first time in the second 

 edition, page 24^8, with this preliminary notice. ''A particular 

 Friend having desired of me that to the preceding Problems I 

 would add one more, I have thought fit to comply with his desire ; 

 the Problem was this." The problem is of no great importance ; 

 it is solved by the method often used in the Ai^s Co7vjectandi, 

 which we have explained in Art. 106. 



273. Problem xiii. relates to the game of Bassette, and 

 Problem XIV. to the game of Pharaon; these problems occupy 

 pages 69 — 82 of the work. We have already sufficiently noticed 

 these games ; see Arts. 154, 168. De Moivre's discussion is the 

 same in all his three editions, except that a paragraph on page 37 

 of the first edition, extending from the words "Those who are ..." 

 to the end of the page, is omitted in the following editions. 

 The paragraph is in fact an easy example of the formulae for the 

 game of Bassette. 



274. Problems XV. to XX. form a connected series. De Moivre 

 solves simple examples in chances and applies his results to esta- 

 blish a Theory of Permutations and Combinations ; in modern 

 times we usually adopt the reverse order, establish the Theory of 

 Permutations and Combinations first, and afterwards apply the 

 theory in the discussion of chances. We will take an example of 

 De Moivre's method from his Problem XV. Suppose there are 

 six things a, h, c, d, e, f, and let two of them be taken at random ; 

 required the chance that a shall stand first, and h second. The 



