AND ON COMBINATIONS AND PERMUTATIONS. 



481 



8. From the foregoing investigation, I deduce the following important Corollary ; that a 

 Elements of various kinds, and singular or plural in their several kinds, will form the same 

 number of different Combinations, whether they combine u, or <j — u, at a time. 



For if we compare the 1st, 2nd, 3rd, &c. terms in formula (xvi.) with the last, penultimate, 

 and ante-penultimate, &c. terms respectively in formula (xviii.) ; and if we make a like comparison, 

 term by term, of the (xvii.) with the (xix.) formula, we shall find that the first term and the last, 

 the 2nd term and the penultimate, the 3rd and the antc-pcnultimate, and so on, are identical, in the 

 series |o, a}, {l, o-}, , \(r — 1, <t\, {<j, <t}, provided it can be shewn that the same law- 

 applies to the terms of each of the two component series, 



i", -r}, {1, -r] {t-1, t}, {t, t}, 



and Jo, a - t\, {l, rr — t}^, {ct-t — 1, cr — t}, \a- - t, <t-x]. 



But when the t, and the (u — t) elements each consist of only one kind, the number of the 

 Combinations that can be formed by taking 0, 1, 2, 3, &c. of these elements at a time, is invariabl\ 

 1, 1, 1, 1, &c. and this series is identical, whether it be taken in direct, or in reverse order. 

 Therefore the law will apply to the series formed by the elements of two single kinds conjoined ; and 

 therefore to the elements of three kinds conjoined ; and therefore universally, of whatever number 

 of different kinds the elements may consist. 



Hence it appears that, to diminish the labour of computation in the application of formulas 

 (xiv.) and (xv.) to particular cases, we ought always to make a selection of the least of the two 

 numbers u and a — u, before substituting one of them for the variable in either of those formulas. 



The theorem just established may also be enunciated in the following terms : 



If the product of any niini1)er of geometrically progressing Polynomes, eadi of which has a 

 limited niunbcr of term.s, and w for the common ratio of the terms, be developed according to the 

 powers of w ; then, assuming a to be the sum of the dimensions of all the Polynome factors, the 

 Coefficient of x°, in the product, will be equal to the Coellicient of x"'". 



9. Ilillierto, the Combinations I have been considering, have been subject only to the condition, 

 ihat they all contained u of the given Elements. Hut we may im])ose tlie further one, that the 



8 Q2 



