?l^8i ARITHMETIC. * 



PROBLEM III. 



Any fiumher of things being ^iven^ whereof there are several 

 given things of one sort^ several of another ^ ^c, to find haiu 

 many changes can be made out of them all. 



RULE.* 



I. Take the scries i, 2, 3, 4, &c. up to the number of 

 things given, and find the product of all the terms. 



2. Take 



* This rule is expressed in terms thus : 



^ — T — I 5 vvhere m = number 



i X 2 X 3,&c.to/ X I X 2 X 3,&c. to ^, &c. 



of things given, p =: number of things of the first sort, q = num- 

 ber of things of the second sort, &c. 



The DEMONSTRATION may be shewn as follows : 



Any two quantities, ^, b, both different, admit of 2 changes ; 

 but if the quantities be the same, or ab become aa, there will be 



but one alternation ; which may be expressed by -Jil_ =1. 



1X2 



Any three quantities, akf all different from each other, afford 

 6 variations ; but if the quantities be all alike, or c^r become aaa^ 

 then the 6 variations will be reduced to i j which may be ex- 



pressed by — — — _lA~i. Again, if two of the quantities only 

 1X2x3 



be alike, or abc become aac, then the six variations will be re- 

 duced to these 3, aacy caa and aca j which may be expressed 



by J2ii2<i=3. 



Any four quantities, ahcd, all different from each other, will 

 admit of 24 variations ; but if the quantities be the same, or 

 al/cd become aaauy the number of variations will be redixced to 



one ; 



