[ 3^° 1 



— -i-. The Prob. that another a is taken next is 2 x — be- 



?2—p fi—p — I 



caufe each ^, bcfides its own chance, has the chance of the b, 

 ■which is derived from it, &c. &c. Whence it follows that the 

 probability that all the a' will be taken before any of the b' is 



t p — I p — 2 I V 2(7 2^6' — I 



n ^ n — I ?; — 2 n — p — r ^ — p " — P — i 



2 X y— 2 X _ . Jl . The probability that the b 



n—p—z n—p + q—i 



derived from the firft a is taken next is : — =, &c. 



* n — p + q 



Whence the prob. tliat -the whole will be taken in the order 

 in which they are written is 



p X ^l - - I X 2 X y X f^^ X f^2 - - J} -j^^ j.^. 

 n. n — I - - - n^p+iq — 3" 



s p q 



ciprocal of which fradlion is the coefficient oi a b . And by the 



p q r s ff 



fame procefs the general coeff. oi ab cd — -a as given in the 

 4>'i law is readily deducible. 



The dem. by the method of permutations is concifely as 

 follows. If the quantities a, a, a, (p) a, b, a, b (iq) were 

 all different, the number of orders is n. n — i. - - i, but 

 as p quantities are the fame, the number muft be reduced by 

 dividing by />./> — i - - i, or the number of permutations 



of 



