DE MOIVRK. 153 



De Moivre treats the subject with great ingenuity and with 

 more generality than his predecessors, as we shall now shew. 



281. Problem xxxv. is thus enunciated : 



Any number of Letters a, h, c, d, e,/, &c., all of them different, 

 being taken promiscuously as it happens : to find the Probability that 

 some of them shall be found in their places according to the rank they 

 obtain in the Alphabet; and that others of them shall at the same time 

 be displaced. 



Let n be the number of the letters ; suppose that j) specified 

 letters are to be in their places, q specified letters out of their 

 places, and the remaining n —p — q letters free from any restric- 

 tion. The chance that this result will happen is 



n{n—l)...(n—j)+l)\ 1 n—p 1.2 (n— p)(?i— ^ — 1) 

 This supposes that p is greater than ; if ^ = 0, the result is 



Iw"^ 1.2 n{n-l) '" 



If we suppose in this formula q=^m — 1, we have a result akeady 

 implicitly given in Art. IGl. 



In demonstrating these formulae De Moivre is content to ex- 

 amine a few simple cases and assume that the law which presents 

 itself will hold universally. We will indicate his method. 



The chance that a is in the first place is - ; the chance that a is 



in the first place, and h in the second place is — , — -:r^ : hence the 



^ ^ n{n—i) 



chance that a is in the first place and h not in the second place is 



1 1 



n n (ii — 1) * 



Similarly the chance that a, h, c are all in their proper places is 

 1 



/ IN / — K^ ; subtract this from the chance that a and h are in 

 n [71 — 1) (n — 2) ' 



their proper places, and we have the chance that a and h are in 

 their proper places, and c not in its proper place : thus this chance is 



1 1 



n 



(n - 1) n («. - 1) {n - 2) 



