loG DE MOIVRE. 



...tlien let all the quantities I, r, s, f,v, &c. be written down with 

 Signs alternately positive and negative, beginning at 1, if]) be = 0; at r, 

 if^9 be = 1; at 5, if p be = 2; &c. Prefix to these Quantities the Co- 

 efficients of a Binomial Power, whose index is = q; this being done, 

 those Quantities taken all together will express the Probability re- 

 quired. 



284. The enunciation and solution of Problem xxxvi. are as 

 follows : 



Any given number of Letters a, h, c, d, e, /, &c., being each repeated 

 a certain number of times, and taken promiscuously as it happens : To 

 find the Probability that of some of those sorts, some one Letter of each 

 may be found in its place, and at the same time, that of some other 

 sorts, no one Letter be found in its place. 



Suppose n be the number of all the Letters, I the number of times 



that each Letter is repeated, and consequently j the whole number of 



Sorts : supj)ose also that p be the number of Sorts of which some one 

 Letter is to be found in its place, and q the number of Sorts of which 

 no one Letter is to be found in its place. Let now the prescriptions 

 given in the preceding Problem be followed in all respects, saving that 



r must here be made = — , s = —, =^- , t — — ; 7T-7 777 , &c., and 



n n {ii — 1) n {n —1) [71 — 2) 



the Solution of any particular case of the Problem will be obtained. 



Thus if it were required to find the Probability that no Letter of any 



sort shall be in its place, the Probability thereof would be expressed by 



the Series 



^ ^"^^ 1.2 ^ 1.2.3 ^ 1.2.3.4 '^'^''• 



of which the number of Terms is equal to q+ 1. 



But in this particular case q would be equal to -j , and therefore, the 

 foregoing Series might be changed into this, viz. 



1 n-l I {n-l){n-2l) 1 {71-I) {n-2l) (n-Sl) 



2 n-l 6 {n - 1) {n - 2) '^ 24: (n-l) {n - 2) (n - 3) '^* 



of which the number of Terms is equal to — j— . 



