480 Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



conjoined; that is, by cr elements, of s different kinds, which combine u at a time; assuming the / 

 kinds to be different from the s — t kinds. 



Let the Combinations which can be formed by 



taking u* at a time of the <j elements, be [ii, a\. 



V T {«, tJ. 



... u-v or - T \it ~ V, ct-t}. 



Imagine some determinate values given to the variables u and «. Every Combination J?« - u, 

 T - t} may be paired with every Combination [v, t] ; and thence will arise ]it — v, <r — t\ 

 X {v, t} different pairs, each containing u elements of the s kinds. If u remains constant, while 

 v varies, there will be a pair for every value of v from to u, if m < t ; and from to x, if u > t. 



Thus we have, for u < t, 



{?/, o-} = {u, a - t} {o, t} + \u - 1, a- - t\ x {l, r] + 



+ {l, (T-t\ {u- i, t\ + {o, a -t\ \u, t} (xvi). 



For M > T, 



{u, <t} = {u, o--t} {o, t\ + \u- 1, ct-t} {l, TJ + (xvn). 



+ {u - T + }, o--t}{t-I, t\ + \u - t, a- - t\ \t, TJf. 



Now in each of these expressions write {a - u) for u ; and we shall have in the former 

 a - u>(T - T ; therefore some of the terms at the commencement of this formula fail ; in the second 

 expression we shall have cr — u < a — t. 



For cr - u > cr - T, 

 I <T - u, cr] = [cr - T, <T ~ t\ \t - u, t} + {ct - t - «., ct-t} \t, t} (xviii). 



For a- - ti< (T - T, 

 {<T - II, (T \ = {cr - u, or - t} {O, t\ + \a - t - u, cr - t| {t, t} (xix). 



I shall apply this method to the two examples before given, where we have for the Combinations 

 "f A, BB, CCC, taken 0, 1, 2, 3, 4, 5, 6 at a time, 



I, 3, 5, 6, 5, 3, 1 Combinations; 

 and for the Combinations of 



DD, EE, taken 0, I, 2, 3, 4 at a time, 



1, 2, 3, 2, 1 Combinations. 



From these numbers, we have to determine the number of the Combinations of all the Elements 

 A, BB, CCC, DD, EE, taken together o, i, 2, 3, i, 5, 6, 7, 8, 9, 10 at a time. 



• 1 write {«, <r} to tleiiote the number of Combinations formed I 1 "> «■ I - ^ef 1 ^'"^ I ^ { M-v. ff — t } J ; 



by ^elements, plural or singular, of any kind or numbers of lands, | ^„j ,^6 terms of the table itself are given by the equation 



when those elements are taken u at a time : ( 1 is "sed instead of . [„,„]= [„, „][„_„, OJ ; which means that the term on line u. 



I J, in order not to confound Combinations with Partitions. column i>, is equal to the product of the term on line v, column 



+ Tile formula is perhaps better given in the condensed form, ! v, by the term on line (w— v), column 0. 



