15^ DE MOIVRE. 



De Moivre uses a peculiar notation for facilitating this process. 

 Let + a denote the chance that a is in its proper place and — a the 

 chance that it is out of it ; let + J denote the chance that h is in 

 its proper place and — h the chance that it is out of it ; and so on. 

 And in general let such a symbol as -\- a-\-h -\- c — d — e denote that 

 a, h, c are in their proper places, and d, e out of theirs. 



n ' n(n -1) ' n{n — l)(n — 2) 



1 ^ 



n(n-l){n-2){n-B)~'^'"' 



Then we have the following results : 



+ h =r 



+ h + a = s 



+ h — a = r—s (1) 



+c+h =s 

 + c-\-h + a = t 



+ c + h-a = s-t •. (2) 



+ c - a =r-s by (1) 



+ c - a + J = s-t by (2) 



+ c-a-b= r-2s + t (3) 



+d+c+h =t 



■Yd+G^-h-{-a = v 



+ d + c-\-h-a = t-v (4) 



+ J+c-a ^s-t by (2) 



+ c?+c — a + &= t — v by (4) 



■\-d-\-c-a-h= s-2t^v (5) 



■\-d-h-a =r-2s-\-t by (3) 



-\-d—h — a-\-c= s-2t-\-v by (5) 



d — h — a — c— r— 35 + 8^—1? (6) 



It is easy to translate into words any of these S3rmbolical pro- 

 cesses. Take for example that which leads to the result (2) : 



