AND ON COMBINATIONS AND PERMUTATIONS. 483 



Since the s kinds of Elements can be combined sr at a time in — L — different ways, and siiire 

 the different s parts of the above partition admit of being permuted, and in that way differently 



distributed among the z kinds of Elements, in — -, —r, — zrTT-r; different ways, the number of the 



o 1 M 1 " 1 



different Combinations of the proposed form, in case of unlimited repetition, will be 



s'\-' r'l' _ s']-' 



js 11 l'"|' l"'!' J'""U ~ 2'" I' |m'l I j""'! 1 V'^"/' 



and if corresponding to every different ss — partition of u, we construct a similar expression, the sum 

 of these will give the total number of the Combinations which can be formed from the s kinds of 

 Elements, when in each Combination there are u Elements of :i kinds. 



11. In the case of unlimited repetition, the aggregate of all the terms, containing u Elements of 

 z kinds, admits of Summation. For, if in each of the z — partitions of the number ?<, the parts be 

 permuted one with another, the number of all these permutations will be 



equivalent terms in the developement of the Binomial [l + l]"~'. This will appear from the 

 following consideration. In the case of > partition, the parts can be permuted in one way. 



In the case of > partition, the parts can be permuted in {u — l) ways. Integrate the 



factorial successively up to 2'"' [l], or 2'"" [u - l] ; and the formula (xxi.) will be the Integral. 



Consequently, the number of the different Combinations, containing u Elements of sr kinds, 

 will be 



-jVp- X ^.-IM (''Xll). 



Example. How many Combinations, containing eight Elements of three kinds each, can be 



4.3.2 7.6 



formed from four kinds of Elements, unlimited in number. Answer — — '- — x — ^ = 84. 



1.2.31.2 



Now the sum of all the terms of the form (xxii.), from z - 1 to z = u, ought to be equal to the 

 g" 1 1 

 Coefficient of w'\ or to -;7|, : and accordingly, if we give to the product (xxii.) the form \«, a} 



X 1^ - 1, ?/ - 1 1 , it will appear to be a particular case of the general theorem, Article 7 of the present 



Section, last demonstrated ; so that S" [f;?, «} x {» - I, ?< - 1 1] = {u, s + u - t\ =-V|;' '''^ ''•'*' 



r ^ I 



term in the developement being 0. 



12. Suppose the Elements in the s given kinds to be limited in point of number. Let it be 

 required to form, from these elements, Combinations, each containing (u) elements of » kinds, with 

 this further limitation, that 



m of the z kinds shall contain v elements, each : 



rn' V : 



m" ,. v" : and so on. 



1st. If none of tiie given kinds contain as many elements as are denoted by any one of the 

 numbers v, n', v", ... , no such Combinations as are required, can be forinud from the given 

 elements. 



