DE MOIVRE. loT 



285. De Moivre then adds some Corollaries. The follo^\ing 

 is the first of them : 



From, hence it follows, that the Probability of one or more Letters, 

 indeterminately taken, being in their places, will be expressed as fol- 

 lows : 



_ 1 iij-l 1 i ll - I) {n - 21) _ j_ {n -l)(n- 21) {n - 3 Q 

 2 n-1^ ^\r-1){ji-2) 24< (n- 1) {n-2) (n-S) 



This agrees with what we have already given from Nicolas 

 Bernoulli ; see Art. 204. 



In the next three Corollaries De Moivre exhibits the pro- 

 bability that two or more letters should be in their places, that 

 three or more should be, and that four or more should be. 



286. The four Corollaries, which we have just noticed, are 

 examples of the most important part of the Problem; this is 

 treated by Laplace, who gives a general formula for the proba- 

 bility that any assigned number of letters or some greater number 

 shall be in their proper places. Theorie. . .des Proh. pages 217 — 222. 

 The part of Problems xxxv. and xxxvi. which' De Moivi-e puts 

 most prominently forward in his enunciations and solutions is 

 the condition that p letters are to be in their places, q out of 

 their places, and n — ij — q free from any restriction ; this part 

 seems peculiar to De MoIato, for we do not find it before his time, 

 nor does it seem to have attracted attention since. 



287. A Remark is given on page 116 which was not in the 

 preceding editions of the Doctrine of Chances. De Moivre shews 

 that the sum of the series 



111 



1 — o + ^ " oT + • • • wi infinitum, 



is equal to unity diminished by the reciprocal of the base of the 

 Napierian logarithms. 



288. The fifth Corollary to Problem xxxvi. is as follows : 



If A and B each holding a Pack of Cards, pull them out at the same 

 time one after another, on condition tliat every time two like Cards are 



