EULER. 245 



from the last term but one we obtain 



2"'"' \m-l-r\r-2' 



and proceeding in this way we find for the sum 



_1 m-l-r |^~TT2 "^^ 17273 ''+-"j 



a^ir 



2»'+i 1^ 



(- 1)''"" I m 



-^ 1 - (1 - 2)' . 



n — 1 — r I ) 



This vanishes if r be an et;e?i number ; and is equal to 



'2r\r \m-l-r ' 

 if r be odd. 



Thus Euler's formula follows from Nicolas Bernoulli's. 



438. The next memoir by Euler is entitled Sur la prohabilite 

 des sequences dans la Lotterie Genoise. This memoir was published 

 in the volume for 1765 of the Histoire de V Acad.... Berlin; the 

 date of publication is 1767; the memoir occupies pages 191 — 230. 



439. In the lottery here considered 90 tickets are numbered 

 consecutively from 1 to 90, and 5 tickets are drawn at random. 

 The question may be asked, what is the chance that two or 

 more consecutive numbers should occur in the drawings? Such 

 a result is called a sequence ; thus, for example, if the numbers 

 drawn are 4, 5, 6, 27, 28, there is a sequence of three and also a 

 sequence of two. Euler considers the question generally. He 

 supposes that there are n tickets numbered consecutively from 1 to 

 n, and he determines the chance of a sequence, if two tickets are 

 drawn, or if three tickets are drawn, and so on, up to the case in 

 which six tickets are drawn. And having successively investigated 

 all these cases he is able to perceive the general laws w^hich would 

 hold in any case. He does not formally demonstrate these laws, 

 but their truth can be inferred from what he has previously given, 

 by the method of induction. 



