Arrangements and Group.s. 125 



number of arrangements divided by | r | ^ | / . . , or the number 

 of ways of making up ?i things into sets, of which the first con- 

 tains r things, the second s things, the third / things, and so on, 

 equals 



|5._ 



II. Given a set of K things, Another set of L things, 

 Another of M things, and so on ; to find the Number of 

 Groups that can be Made by taking r Things from the 

 First set, s Things from the Second set, t from the Third 



SET, AND so on. 



IK 

 Of the K things taken r at a time there are , — — j groups, 



" 1 L "" 

 and of the L things taken 6- at a time there are p , y - groups, 



s Lt s 



\U 

 and of the M things taken / at a time there are f^-i^ , groups, 



and so on, and as any one of the groups from the first set may be 

 taken with any one of the groups from the second set, and any 

 one from the third set, and so on, to form a larger group, it follows 

 that the total number of these larger groups equals the product 



I K I L 1 M 



|r|K-r \s_ \ L- s \J [ M- / 



12. There are various relations connecting arrangements with 

 arrangements, groups with groups, arrangements with groups, 

 etc. We will obtain a few of these relations, and recommend 

 that the student try to obtain others not. here given. 



One relation was obtained in Art. 2, where it was shown that 



A(;:)=;/AC;ii), (I) 



and another in Art. 3, where it was shown that 



AC;)=^/Aa'zl). (2) 



We have already found that 



A(';)=,li, (s) 



