AND ON COMBINATIONS AND PERMUTATIONS. 485 



How many Combinations can be formed in accordance with that type ? 



Here s = 4 ; z = 3; m = \ ; m' = 2. 



4.3.2 



And the number of Combinations = — -t- ,- = 12. 



I'I'.l'l' 



Given the same elements. — How many Combinations can be formed in accordance with the type 

 6, 1, 1 ? 



Since 6 is greater than the given limit 5, the Answer is 0. 



14. Before closing this Section on Combinations, I shall beg to notice that all of the theorems 

 it contains, admit of an important application, and that is, to the properties of Composite Numbers. 



It is known, for instance, that, if the elements J, B, C, &c. represent primes, a composite 

 number, of the form A". B^. C. &c., will liave the total number of its divisors represented by (a + 1) 

 (/3 + 1) (7 + 1) &c.; but if the question be, how many divisors such a number has that are of u 

 dimensions, the answer to that question will be obtained by means of formula (xiv). But it will 

 suffice to have hinted at these analogies. 



k 



SECTION III. 

 On Permutations. 



1. When there are s different kinds, each containing only a single element, these elements 

 iken M at a time, will form «° | "' different Permutations ; where s° | "' = 1" | ' x the coefficient of .r" 



n [1 + ,i]~ developed. But when any of the « kinds contain more than one element, and the 

 plurality of the elements is short of infinity, it is only in the particular case where u is equal to the 

 united number of all the elements belonging to the s kinds, that the number of the permutations 

 has hitlierto been determined. In this case, if there be a elements of one kind, |8 of a second kind, 

 7 of a third kind, &c., and a + /?{ + 7 + &c. = o-, the number of the permutations formed by the a 



things taken all at a time, is — 7- — s-rr n , according to the well-known theorem. 



° 1" ' . 1**! ' . i^l'.&c. " 



2. The latter formula denotes the number of permutations which the a elements of the kind A, 

 the /3. elements of the kind B, the 7 elements of the kind C, &c. are capable of forming, when, 

 instead of being permuted indiscriminately, the .^'s, the fi's, the Cs, &c. change their order of 

 sequence in resjiect of one another, but in respect of the elements of their own several kinds, preserve 

 an immutable order of sequence. If the a elements A, not to the full extent of 1"| ', but to some 

 limited extent, undergo the permutations P{a) ; and in like manner the /3 elements B to the limited 

 extent P(fi)-, and the 7 elements C to the limited extent ^(7), &c. the number of the permutations 

 which the cr things will then together form, will be 



i'iM^i'Tv|',&c . ^ ^^"^' ■^^^^' ^^y^' ^' ^"''"^- 



3. To determine generally the number of permutations which can be formed from any given 

 set of elements, taken u at a time. 



Let any partition of u be p + q + r + &c. = v. 



It has been shewn, in Articles 12 and 13 of the preceding Section, how to determine the number of 

 I all the Comljinations which can be formed from a given set of elements, when each Com1>ination is ti> 

 I consist of u elements of ;:; kinds, and i.s to accord with any particular partition of u, or type. If 



