CHAPTER IX. 



ARRANGEMENTS AND GROUPS. 



I . Definitions. Every different order in which given things 

 can be placed is called an Arrarigement or PermiUation, and every 

 different selection that can be made is called a Group or 

 Combination. 



Thus if we take the letters a, b, c two at a time there are six 

 arrangements, viz : 



ab, ac, ba, be, ca, cb, 



but there are only three groups, viz : 



ab, ac, be. 



If we take the letters a, b, e all at a time, there are six 

 arrangements, viz : 



abe, aeb, bae, bea, cab, eba, 

 but there is only one group, viz : 



abe. 



2. PROBI.EM. To FIND THE NUMBER OF ARRANGEMENTS OF 



n Different Things taken All at a time. 



First. If we take one thing, say the letter a, there can be but 

 one arrangement, viz : the thing itself. 



Seeond. If we take two things, say the letters a and b, there 

 are two arrangements, viz : 



ab, ba. 



Third. If we take three things, say a, b, r, there are six 

 arrangements, viz : 



abe, aeb, bae, bca, eab, eba. 



Notice that there are two arrangements in which a stands first, 

 two more in which b stands first, and two more in which c stands 

 first. 



Fourth. If we take four things, say a. b, e, d, then we may 

 arrange the three letters b,-e, d in every possible way and place a 

 before each arrangement, then arrange the three letters a, c, d in 

 every possible way and place b before each arrangement, then 

 arrange the three letters a, b, d in every possible way and place 



