AND ON COMBINATIONS AND PERMUTATIONS. 477 



If 1 be now substituted for each of the elements A, B, C, &c., the polynomes will respectively, 

 become [l + w + or + ,i"], [l + j? + x^ + .r^], &c., and the coefficient of x" in the de- 

 veloped product, now that 1 has become the value also of each term of the form A^B^C in that 

 product, will represent, not only tlie sum, but also the number of all such terms : that is to say, 

 of the different combinations which can be formed with the a elements, taken u at a time. 



•1. That coefficient is an explicit function of u, which I now proceed to determine. 

 The product of these geometrical polynomes, is 



1 - ^''+' 1 - .v^*' 

 . . &c. : 



\ — X \ — X 



that is, (1 -.r)-' [l ->?;''+'] [l -.r^ + '] .&c (xi.) 



But [l - ,r]-'= 1 +«a;+ -^ ^v" + 7^''^" + 



= r^ [!'- ' I ' + 2«-' I ' *■ + [m + l]'-' I ' w" +]. 



For the sake of brevity, 



Let u+l be represented by u/-, 



a + I by "/ ' 



/3+ 1 by/3,; &c. 



1st. When each of the s kinds of elements. A, B, C, Sec. admits of unlimited repetition, the 

 required coefficient of x", will be 



I 





and in this case, of plural elements, all kinds admitting of unlimited repetition, a solution of the 

 comljination problem, to the same effect as the preceding, has, as Mr. De Morgan informs me, 

 been given by Hirsch. 



adly. When the elements of one kind, A, are limited in number to a, but the elements of 

 the other (s - 1) kinds may be repeated without limit, the required coefficient, (which is that of 

 (1 -.r)-'[l -a;"]), will manifestly be 



P^[";-r-K-«J-T]' (''"), 



from which expression however, the second term is to be excluded, in case [m, - aj should be 

 negative. 



.3dly. When the elements of two kinds, A and B, are limited in number to a and fi re- 

 spectively, but the elements of the other s - 2 kinds may be repeated without limit, the required 

 coefficient, (which is that of .r " in the developement of (1 - .r)"' [l - .r"] [l - .t"]), will be obtained 

 by performing on formula (xn) with fi the same operation that was before performed on formula 

 (xi*) with a . Tile result will manifestly be 



, («;-'!'- [«--«J"T+[",-«.-/3,l-r I 



-II 



(xm). 



• I line the facloriiil notatidii, in which 



»"| ' rcprcpicnm « («H- I) (s +2) U + ("~^)i' 



and «"|-i «(»-l)(»-2) f«-(u-l)]. 



