50 



Consequently the required sum is 



li]T X p-i|i ( XVL ) 



If in (XVI.) z varies from to u — 1, 



this summation being a particular case of formula (XI.). The result 

 agrees with D M [1] formula (IX.), art. 3. 



10. When the given elements are all finite in number, we may 

 determine {m, erj, by taking the sum of all the particular determina- 

 tions that may be obtained pursuant to art. 9, by giving to z the 

 successive values 0, 1,2, 3, &c. If u < s, the upper limit of z is u, 

 and the number of types to be formed is [2w, u{]; which becomes 

 [2s, s^] , if w=s. If u > s, the upper limit of z is s; and the number 

 of types to be formed is [m + s,^]. (See articles 4 and 5, Section I.) 

 But, if the repetition is finite, some of these partitions may fail to 

 yield combinations. 



11. If the elements A, B, C, &c. represent different prime num- 

 bers, all the methods and theorems contained in this section will 

 apply, mutatis mutandis, to the composite numbers of which those 

 primes, or the powers of those primes, are divisors. 



* 

 III. On Permutations. 



1. Let the given elements be of s different kinds. We can de- 

 termine in two known cases, by an explicit function of u, when the 

 elements are taken u at a time, in how many different ways they can 

 be permuted. The number of the permutations is denoted, when 

 there is but one element of a kind, by s m|_1 ; and when in all the 

 kinds the elements are plural without limit, by s u . When the plu- 

 rality is finite, it is only in the particular case of all the elements 

 being permuted at a time, that there is a known formula to express 

 the number of their permutations. 



2. Every combination constructed on a given type, u=mv + m'v' 

 + m"v"-\- &c, will generate the same number of permutations, 



_H± =p 



rpliiTO rp'iiim'rjyiiTm" _ g, c _ 



Therefore, if the number of the different combinations which can be 

 constructed out of the given elements in conformity with that type, 

 is represented by Q, Q x P will be the number of the permutations 

 corresponding to the type and to those elements. If the plurality 

 be without limit, 



lmll. 1»»'[1 < p»"|i # ^ c> 



xP 



will be the number of the permutations. If the given elements be 

 finite in number, as in formulas (XIV.) and (XV.), the number of 



