AND ON COMBINATIONS AND PERMUTATIONS. 489 



The series may be expressed by 



„/M=«|-'.s"-«|-'\ 



n 



the limits of 0, when n is even, being and — ; 



■,1*- \ 

 odd and 



2 



This gives the number of permutations formed by « dual sets of elements, when taken u at 

 a time. 



Thus when m = 6, we have 



6'|-'s'|-i eM-'.s"!-' 6" 



-1 C.3 -I 



+ ' „ ' + — ! 1 + 



2 2.4 2.4.6 



And when u — 7, 



rye I - 1 „4 I - 1 



+ „ — + — L_ + 



7" 

 s"-' ■ ■ ■ ■ ■ 



2.4 2.4.6 



When u = 6, let s = 3. 



6.5.4.3.2.1 X 3.2.1 



The number of the permutations 



2.4.6 

 = 5x3x3x2 

 = 90. 

 When u = 7, let s = 4. 



7.6,5.4.3,2,4,3.2.1 

 The number of permutations is then = — — '- — '- — '- — '- — '- — '■ — '- — '— = 7.6.5.4.3 = 2520, 

 ^ 2.4.6 



It is not improbable that a further developement of the series obtained by Arbogast's method, 

 and a subsequent equating of particular terms to 0, might lead to other symmetrical and curious 

 results. 



8. In Art. 7 of Section II., I have given a method, from the known combinations of two 

 independent and separate sets of elements, differing from one another in kind, to determine the 

 combinations of the two sets, when united. I proceed to apply a precisely similar method to Per- 

 mutations. 



If « of T elements form \v, t\ combinations, and (m — v) of (cr — x) elements form [u — v, 

 a — t\ combinations, they will together form 



ju, t] X ^u — V, cr - t] combinations. 

 Take some one of the {v, t] and some one of the [u —v, a — t] combinations. If it be a 

 condition that the elements belonging to these two sets, separately considered, shall preserve their 

 original order of sequence, but that those of one set of elements, as compared with those of the other, 

 may change their places, the permutations formed by the two sets of this particular combination 

 united, will be 



1"|' 

 i"|! i"-"|i' 

 But, if the V elements, and the u - v elements, considered separately, may change their places, 

 and the former may undergo P{v) changes, and the latter P(_u - o) changes, and if the same may 

 be predicated of the elements contained in each different pair of combinations, \v, t\ and {u - i», 



<T - t}, 



3 n2 



