484 Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



2ndly. If any of the given kinds contain fewer elements than are denoted by the least of the 

 numbers v, v\ v" ... , such kinds, it is manifest, may be wholly omitted from consideration. 



Srdly. If any of the given kinds contain more elements than are denoted by the greatest of the 

 numbers v, v', v" ... , the excess above such greatest number may be wholly omitted from con- 

 sideration ; and, in the same manner, if any of the given kinds contain a number of elements 

 intermediate between two of the numbers v, v', v", ... , the excess above the least of these two 

 numbers may, in the course of the operation hereinafter directed, be wholly omitted from con- 

 sideration. 



Thus the given set of elements admits of reduction to t kinds, containing at least v elements 

 each : 



-I- T' kinds, none of which contain v elements, but each of which contains at least v' elements : 



-I- T' kinds, none of which contain v' elements, but each of which contain at least t>" elements, &c. 



Thus there will be t kinds to supply m kinds in each Combination with v elements each : 

 / ^ m + T' = t' kinds, to supply m' kinds in each Combination with v' elements, each : t' - m' 

 + T' = t" kinds, to supply m" kinds in each combination with v" elements each : and so on. 



Therefore since m kinds have been chosen out of t kinds; 



m out of t' ... ; 



m' out of t" ... ; &c. 



the number of the Combinations of kinds that will be formed, in which the several kinds will 

 contain the requisite number of elements, will be 



— ^,-r X -— 7rT X -— .. - X &EC (xxiu). 



1'" I ' 1"' I ' 1'" I ' 



Example. — From the elements F\ E", D\ (7, JS", A, how many Combinations of the form 

 or type, 



s " 1 1 



.5, .., i, 1 



can be constructed ? 



Since 3 is the highest number in the type, reduce the given elements from 



6, 5, 4, 3, 2, 1, 



90. 



13. When, however, after previous reduction, if requisite, the limited number of elements is 

 ihe same in each of the given kinds, and it is required to determine how many Combinations can 

 1)0 formed from those elements in accordance with a given type, either all or none of the kinds 

 will contain the number of elements requisite to form any required Combination : and the formula 

 applicable to the case of unlimited repetition, viz. (xx.) 



'.1'" '. 1" 



is to be applied. 



Example. — Given the elements A°, B', C^, £)'. 



and the type 4, 2, 2. 



