49 • 



than v elements ; each kind in class 2 contains fewer than v, but not 

 fewer than v' elements ; each kind in class 3 contains fewer than v' , 

 but not fewer than v" elements ; and so on ; and so that the given 

 kinds may in this way be reduced, say, to t kinds containing v ele- 

 ments each + T' kinds containing v' elements each + T'' kinds con- 

 taining v" elements each, &c. Then let t — m + T' = t'; t' — m' + T' = t"; 

 and so on. The given kinds being thus ordered, since we are required 

 to select, 1st, m out of t kinds ; then, 2nd, m' out of t' kinds ; then, 

 3rd, m" out of t" kinds ; and so on ; the number of the different 

 combinations which can be constructed from those kinds in confor- 

 mity with the type, will be 



fm\-l lfm'\ — l "fm"\ — l 



LI— . 1— — l „ , &c (xiv.) 



jmll \m'\l |m"ll v J 



If tv + T'v' + T'v". . &c. is reduced to a single term, t.v; then 

 formula (XIV.) becomes 



&c (XV.) 



\m\l w \m'\l t \m"\l. 



Example of (XIV.). Given eight elements of 1 kind, seven of a 

 2nd kind, six of a 3rd, five of a 4th, four of a 5th, three of a 6th, 

 two of a 7th, and one element of an 8th kind, out of which it is re- 

 quired to construct combinations, each consisting of three kinds with 

 five elements each + two kinds with three elements each + one 

 kind with two elements. Of such combinations there can be formed 



43j-i 3*1-1 2M-i _ 94 

 p|i p|i * \\\\ 



9. If it be required to determine how many different combinations 

 can be constructed, each containing u elements of z kinds, and the 

 given elements are all finite in number ; we must form all the differ- 

 ent ^-partitions of u ; and each of these partitions being regarded as 

 a type, we must determine, by formula (XIV.) or (XV.), how many 

 combinations correspond to each of these types ; and the total num- 

 ber required will be the sum of all these particular determinations. 

 But if the given elements may all be repeated without limit, it 

 follows from formula (XIII.), that the sum of all the particular de- 

 terminations may be represented by 



**'-' x jg/ I" 11 \ 



\z\\ ' \\m\\ % \m'\\_ \m"\\ t & C .J" 



Now 



I \m\\^ \m'\\_ \m"\l I 



denotes how many different permutations can be formed, when, in 

 each different z-partition of u, the parts are permuted z together at 

 a time ; and the number of such permutations is 



y- l [i]=y- a [«-i] = [M ~|, 3 _*"; i 1 '" 1 -. 



