ESSAYS 299 



PROBLEMS OF ARRANGEMENT OP AN INFINITE CLASS. 



(Philip E. B. Jourdain, M.A.). 



If we consider a class formed by three different members a, b, and c, we find that 

 we can arrange them in six ways : 



abc, acb, bac, bca, cab, cba. 



These various ways are known as " permutations three at a time," and, in general, 

 the number of permutations of any finite number («) of members of a class, n at a 

 time, is n ! or " factorial n." It is easy to see that the total number of such per- 

 mutations of the various classes of first 1 and then 2 and then 3, . . . , members 

 selected from a class of n members is 



n + n(n- i) + n(n-i)(n-2) + . . . + n(n-i)(n-2) ...2.1. 



For the first term is the number of ways we can select one member out of the 

 n members, the second term is the number of ways we can select two members 

 out of the n members if we regard, for example, ab as a different selection from ba ; 

 and so on. 



Among these permutations certain ones thus consist of the same members, and 

 they merely differ in the order given to these members. If we disregard the 

 various orders, we get " combinations " ; thus bac is a different permutation from 

 bca, but is the same " combination." The number of combinations of n members 

 rata time is obviously less than that of the analogous permutations, and we easily 

 find that the total number of combinations of 1, 2, . . . , n members from the given 

 class of n members — that is to say, the total number of sub-classes — is 



t . *(»-!) , n{n- i)(»-2) , , «(«-!)■• . 3 , ... t 



n+ ~^r~+ ji + •••+ („_ 3) ! +»+ u 



This last sum is easily seen to be the binomial expansion of (1 + 1)"— 1 or a"- 1. 



When we come to consider classes of an infinity of members, such as the class 

 of all integer numbers or the class of all rational numbers between o and 1, we 

 enter the theory of " transfinite numbers," of which the foundations were laid 

 by Georg Cantor. In the first place, the distinction between a " class " of members 

 and a "series" of members then becomes of very much greater importance than it 

 is in the case of finite classes. When we speak of a "class" of members, we do 

 not imply any particular order among the members, whether or no a particular 

 member x " precedes " another member y in virtue of some quality like size or 

 position or colour or loudness of tone. If a class is put in a certain order, whether 

 or not this order is linear or multi-dimensional, we have what is called a " series.'' 

 The series used in mathematical analysis are generally made up by adding or 

 multiplying real or complex numbers so as to form a sequence .Si, .Sj, St, . ■ . , 

 S n , . . . in an order like that of magnitude of what we call the "natural " numbers 

 1, 2, 3, ...,«,... ; but, in the extended use of the word, a " series " can be in 

 any order : thus the rational numbers form a 5< series," although no single term of 

 it has an immediate predecessor or an immediate successor. 



Now Cantor, about 1882, was led by severe meditations on such questions as 

 that, if we represent values of a variable approaching a limit by points X\ t x%, 

 . . ., x H , on a straight line, there is some reason for " numerating " the limiting 

 point to which all the x's tend by another x to which is affixed a suffix (») which 

 we consider as representing not a number just as any one of the previous suffixes 

 does, but a new, non-finite number which is the least number greater than all the 

 'finite numbers n which are used up for the suffixes of the .^.representing the 



