MONTMORT. 85 



ever shews by tlie evidence of a letter addressed to John Bernoulli, 

 dated 15th November, 1710, that he was himself acquainted with 

 the formula before it was published by De Moivi-e ; see Montmort, 

 page 307. De Moivre first published his demonstration in his 

 Miscellanea Analytica, 1730, where he ably replied to the asser- 

 tion that the formula had been derived from the first edition of 

 Montmort's work ; see Miscellama Analytica, pages 191 — 197. 

 De Moivre's demonstration is the same as that which we have 

 given. 



150. Montmort then proceeds to a more difficult question. 

 Suppose we have three sets of cards, each set containing ten cards 

 marked with the numbers 1, 2, . . . 10. If three cards are taken 

 out of the thirty, it is required to find in how many ways the 

 sum of the numbers on the cards will amount to an assigned 

 number. 



In this problem the assigned number may arise (1) from three 

 cards no two of which are of the same set, (2) from three cards 

 two of which are of one set and the third of another set, (3) from 

 three cards all of the same set. The first case is treated in the 

 problem, Article 148; the other two cases are new. 



Montmort here gives no general solution; he only shews how a 

 table may be made registering all the required results. 



He sums up thus, page 62 : Cette methode est un peu longue, 

 mais j'ai de la peine a croire qu'on puisse en trouver une plus 

 courte. 



The problem discussed here by Montmort may be stated thus : 

 We require the number of solutions of the equation x -\- y + z = p, 

 under the restriction that x, y, z shall be positive integers lying 

 between 1 and 10 inclusive, and p a positive integer wdiich has an 

 assigned value lying between 3 and 30 inclusive. 



151. In his pages 63 — 72 Montmort discusses a problem in 

 the summation of series. We should now enunciate it as a general 

 question of Finite Differences : to find the sum of any assigned 

 number of terms of a series in which the Finite Differences of a 

 certain order are zero. 



In modern notation, let iin denote the n^^ term and suppose 

 that the {in + 1)*^ Finite Difference is zero. 



