DE MOIYRE. 155 



this means that the chance that c and h are in their proper places 

 is s ; and this we know to be true ; 



-\- c-\-'b-\- a = t, 



this means that the chance that c, h, a are all in their proper 

 places is t ; and this we know to be true. 



From these two results we deduce that the chance that c and h 

 are in their proper places, and a out of its place is 5 — ^ ; and this 

 is expressed symbolically thus, 



-{-c-\-h — a = s — t 



Similarly, to obtain the result (8) ; we know from the result (1) 

 that r — 5 is the chance that c is in its proper place, and a out of 

 its proper place ; and we know from the result (2) that 5 — /5 is the 

 chance that c and h are in their proper places, and a out of its pro- 

 per place ; hence we infer that the chance that c is in its proper 

 place, and a and h out of their proper places is r — 2s + ^ ; and this 

 result is expressed symbolically thus, 



282. De Moivre refers in his Preface to this process in the fol- 

 lowing terms : 



111 the 3oth and 36th Problems, I explain a new sort of Algebra, 

 whereby some Questions relating to Combinations are solved by so easy 

 a Process, that their Solution is made in some measure an immediate 

 consequence of the Method of Notation. I will not pretend to say that 

 this new Algebra is absolutely necessary to the Solving of those Ques- 

 tions which I make to depend on it, since it appears that Mr. Montmort, 

 Author of the Analyse cles Jeux de Hazard, and Mr. Nicholas Bernoulli 

 have solved, by another Method, many of the cases therein proposed : 

 But I hope I shall not be thought guilty of too much Confidence, if 

 I assure the Reader, that the Method I have followed has a degree of 

 Simplicity, not to say of Generality, which will hardly be attained by 

 any other Steps than by those I have taken. 



283. De Moi^Te himself enunciates his result verbally ; it is of 

 course equivalent to the formula which we have given in Art. 281, 

 but it will be convenient to reproduce it. The notation being that 

 already explained, he says, 



