124 Algebra. 



turbed, there could be twelve arrangements made from this one 



divisions into sets. The twelve arrangements are: 



ab cde ba cde 



ab ced ba ced 



ab dee ba dee 



ab dee ba dee 



ab eed ba eed 



ab ede ba ede 



From any other way of dividing into sets there could be twelve 



arrangements found, hence the whole number of arrangements of 



five letters equals twelve times the number of ways of dividing 



into sets, or the number of sets equals one- twelfth the number 



of arrangements. The number of arrangements in this case is 15, 



hence the number of ways of makimg up sets in this case equals 



tV |i=io. 



We will now take the general case of n letters a, b, e, d, e, f, . . 

 and take the first r letters to form the first set, the following 5 let- 

 ters to form the second set, the next following / letters for the 

 third set and so on. 



Place the letters of the first set down in a horizontal line, then 

 those of the second set in the same horizontal Xxn^ following the 

 first set, and those of the third set in the same horizontal line fol- 

 lowing those of the second set, and so on. 



We thus have all the n letters arranged in a horizontal line, and 

 it is evident that we could keep these sets undisturbed, but still 

 make several arrangements of the 71 letters in a horizontal line. 

 The letters in the first set can be arranged in j r ways, those of 

 the second set in | ^ ways, those of the third set in | t ways, and 

 so on, and as any arrangement in any set may accompany ayiy 

 arrangement in any other set, hence the whole number of arrange- 

 ments while the sets are undisturbed is equal to | r I ^ | / . . . 



Thus from one way of making up the sets there are '■ r\ s\ t . . 

 arrangements, and of course from any other division into sets, 

 there could be formed the same number of arrangements^ hence 

 the whole number of arrangements of n things, all at a time, 

 equals \r\s\t. . . times the number of ways of making up the 

 sets, or the number of ways of making up the sets, equals the 



