254 EULEE. 



tickets shall be drawn. This formula corresponds with Laplace's. 

 We have only to put 771 = w in Art. 453. 



Euler then considers the question in which n — 1, or ?i — 2, ... 

 tickets at least are to be drawn. He discusses successively the 

 first case and the second case briefly, and he enunciates his 

 general result. This is tlie following ; suppose we require that 

 71 — V tickets at least shall be drawn, then the number of favour- 

 able cases is 



+ {i> + l)<j> {», v+2){,j>{n-v- 2, r)}' 



- i^^H^-±^ ^ („, ^ + 3){ct,in-v- 3, r) }'- . . . 



This result constitutes the addition which Euler contributes to 

 what had been known before. 



459. Euler's method requires close attention in order to gain 

 confidence in its accuracy ; it resembles that which is employed 

 in treatises on Algebra, to shew how many integers there are 

 which are less than a given number and prime to it. We will give 

 another demonstration of the result which will be found easier 

 to follow. 



The number of ways in which exactly m tickets are drawn 

 is (^ (n, m) A'" {</) (0, r)Y. For the factor A"^ [(\> (0, r)Y is, by 

 Art. 454, the number of ways in which in a lottery of m tickets, 

 all the tickets will appear in the course of x drawings ; and 

 (n, 77i) is the number of combinations of ii things taken m at 

 a time. 



The number of ways in which n — v tickets at least will appear, 

 will therefore be given by the formula S </> {ii, m) A"* {^ (0, r)Y, 

 where S refers to m, and m is to have all values between n and 

 n — v, both inclusive. 



Thus we get 



A" [^ (0, r)Y + « A'- [</> (0, r)r + ^^^-^ A»-[<^ (0, ryf 

 the series extending^ to i^ + 1 terms. 



