486 



Mr. VVARBURTOxV, ON THE PARTITION OF NUMBERS. 



that partition be p, q, r, then, in the number of combinations so determined, are included, not 

 only those of the form J''B'C\ VET, &c. (in which the kinds of elements are changed), but 

 also those of the form A'BfO, D'E'Ff, &c. (in which the kinds remaining the same, the order of 

 sequence in the numbers p, q, r is altered). Let the total number of the combinations correspondmg 

 to one such partition of u be denoted by Q. Then since every such combination wiU give rise to 



'''^'^'^ = ! different permutations, if we denote -r „ , i ,,. .r.i ^y -P' 



iPp I'l'.rl'. l''|'.l'l'.l'l' f] .I'l .1 1 



every different partition of u, or type, will give rise to Q x P permutations. We must therefore 

 determine by Articles 13 and 14, the number of the combinations corresponding to all the different 

 partitions of u, and also the corresponding permutation factors, and take the product ; and the sum 

 of all these particular products, or ^•[Q x P], will give the total of the permutations which can be 

 formed from the elements taken u at a time. 



1st Example. Given the set of Elements^*, B', C°. Required the number of all the Com- 

 binations and Permutations of those Elements, when 7 are taken at a time. 



Here i< = 7 ; « = 3 ; and, since all parts are to be excluded which exceed 4, s: in this case 

 aries only from 2 to 3. 



2nd Example. Given the set of Elements J\ B'\ C'' ; required the number of all the Com- 

 binations and Permutations of these Elements, when 8 are taken at a time. 



Here m = 8 ; s = 3 ; and, since all parts are to be excluded which exceed 5, z in this case varies 

 only from 2 to 3. The combinations are here obtained by the formula 



l" 



Given 



Elements. 



Parti- 

 tions of 8 



.1" 



5, 5, 5 



5, 3, 



4, 4, 



5, 2, 1 

 4, 3, 1 

 4, 2, 2 

 3, 3, 2 



Combina- 

 tions. 



T^ = 6 



I . a ^ 



3.2.1 _ Q 



r~T~2 — ^ 

 M4 = 3 



Permutation 

 Factor. 



27 



No. of Per- 



mutationti. 



336 

 210 

 1008 

 1680 

 1260 

 1680 



6174 



4. The method I have just described, of determining in succession the permutations corre- 

 sponding to the different partitions of the number ii, must, in cases of limited repetition, have 

 been adopted to determine also the number of the combinations, when ti elements are taken at a 



