'252 EULER. 



number of ways is the number of permutations of n things taken 

 all together. 



Thus we see that the sum of a certain series might be inferred 

 indirectly by the aid of the Theory of Probability ; we shall 

 hereafter have a similar example. 



452. In the Memoires ... par divers Savans, Vol. VI., 1775, 

 page 363, Laplace solves the following problem : A lottery con- 

 sists of n tickets, of which r are drawn at each time ; find the 

 probability that after x drawings, all the numbers will have been 

 drawn. 



The numbers are supposed to be replaced after each drawing. 



Laplace's method is substantially the same as is given in his 

 Theorie . . . des Prob., page 192; but the approximate numerical 

 calculations which occupy pages 193 — 201 of the latter work do 

 not occur in the memoir. 



Laplace solves the problem more generally than he enunciates 

 it ; for he finds the probability that after x drawings m specified 

 tickets will all have been drawn, and then by putting n for m, 

 the result for the particular case which is enunciated is obtained. 



453. The most interesting point to observe is that the pro- 

 blem treated by Laplace is really coincident with that treated by 

 De Moivre, and the methods of the two mathematicians are sub- 

 stantially the same. 



In De Moivre's problem 7i^ is the whole number of cases ; the 

 corresponding number in Laplace's problem is [^ (n, r)}'', where 

 by (/) (71, r) we denote the number of combinations of n things 

 taken r at a time. In De Moivre's problem (n — ly is the whole 

 number of cases that would exist if one face of the die were 

 expunged ; the corresponding number in Laplace's problem is 

 j^(7i-l, r)]^ Similarly to (n — 2y in De Moivre's problem 

 corresponds [(f) (n — 2, r)]"^ in Laplace's. And so on. Hence, in 

 Laplace's problem, the number of cases in which m specified 

 tickets will be drawn is 



{<P (n, r)Y-m {4, («- 1, r)}' + "^"~^^ (<^ (»- 2, r)}' - ... ; 



and the probability will be found by dividing this number by the 

 whole number of cases, that is by {</> (?i, r)}^ 



