COMBINATIONS AND PERMUTATIONS. 



215 



rith negative terms. I may arrange the combinations as 

 follows : 



ABCD . Four out of four . . i combination. 



ABCd 

 ABcD 



AZ>CD 

 BCD 



Three out of four . . 4 combinations. 



A6cD 

 AbCd 



a&CD 



Two out of four 



6 combinations. 



Abed 



. 7 

 abCd 



abcD 



One out of four ... 4 combinations. 

 abed . . None out of four . . i combination. 



The numbers, it will be noticed, are exactly the same 



those in the fifth line of the arithmetical triangle, and 

 in exactly similar correspondence would be found to 

 jxist in the case of each other column of the Abece- 

 larium. 



Numerical abstraction, it has been asserted, consists in 

 >verlooking the kind of difference, and retaining only a 

 consciousness of its existence (p. 177). While in logic, 

 then, we have to deal with each combination as a separate 

 :ind of thing, in arithmetic we can distinguish only the 

 classes which depend upon more or less positive terms 

 )eing present, and the numbers of these classes imme- 

 liately produce the numbers of the arithmetical triangle. 



It may here be pointed out that there are two modes 



