AND ON COMBINATIONS AND PERMUTATIONS. 487 



time, had not the formulee xiv and xv, or the metliod described in Article (6) of the former Section, 

 afforded a readier method of attaining the same object. In the case of Permutations, I have not 

 succeeded, except in certain cases, in readily determining by means of an explicit function of u 

 the number of the permutations formed by m elements of s kinds. 



5. It is a known formula, that when the elements in all the s kinds admit of unlimited repe- 

 tition, the number of the permutations which can be formed by taking m elements at a time, is 

 expressed by «°. 



If we take the form I have assumed, in Articles (10) and (12) of the former Section, for the 

 resolution of u into z parts, where these parts are represented by 



v, V, V, ... (tn) v', v', v', ... (m) v", v", v", ... (m"), 



and m + m' + m" + = a, 



and mv + in v + iri' v" + = m; 



we shall, in the case of unlimited repetition, have the combination factor, 



1--|' 



^' ~ ym ll im' I 1 \^"\ 1 ' 



and the permutation factor, 



•^' r I'l'i'" rr'i n*"' ri""|n'n" ^xxv;. 



the partitions being those in which the lower limit of the parts is 1 ; and % extending from 1 to m 

 when u < s\ and from 1 to «; when ?/ = or > s. 



But, by Art. 7, Sect. I, p. 5, [«, 1,] + [m, 2i] + [m, x;\ = [?<, s-J ; 



and the product Q ^ P therefore coincides with the expression given by Lagrange, in his demon- 

 stration of the Polynome theorem, for the terminus generalis of the expansion of 

 ^[i+i+i...i.)> or £- X ^^^ ^. (j)^ 



when multiplied by the factor l"|'. The terminus generalis, so multiplied, is 



/ 1''. 1' 1' \ 



[1 + 1 + 1 + ...(«)]" = !" '.y -n — , ,;•• 1, 



\i^ . r' . 1' . .../ 



where p, q, r, &c., are all the different parts obtained by s - partitioning u, the lower limit of the 

 parts being ; and for every determinate set of values assigned to p, q, r, &c., these letters 

 receiving every different order of sequence possible. 



In the case, therefore, of unlimited repetition, the number of permutations which can be formed 

 by taking u elements of s kinds at a time, is the coefficient of a;" in the product of the s infinite 

 series, 



\_l + X + + &c.] [l + a? 4- + &c.] (s) ; multiplied by l"|'. 



6. It is manifest, therefore, that if with respect to the elements of any one kind, A, we restrict 

 the number of elements to a ; and in another kind, B,tofi; and so on, we must make a correspond- 

 ing restriction in the terms of 1, 2, or more, of the above Polynomes. And this leads to the followiiifj 

 theorem : viz. that the number of the permutations which can be formed by the elements of s kinds, 

 whose respective limits are a, /3, 7, &c., when those elements are taken u at a time, is the 

 coefficient of «" in the product of the s Polynomes, 



(^1 + 3? + — —Tinj ('+^ + 772 ■*■ — Fpj'^'^-' ">">"?'"-■'• ^y ' I (""^'O' 



Vol. VIII. Part IV. 3 II 



