226 ARITHMETIC* 



PROBLEM VII. 



31? jfind the compositions of any numhery vi an equal number of 

 setSy the things themselves being all different. 



RULE.* 



Multiply the number of things in every set continually 

 together, and the product will be the answer required. 



EXAMPLES. 



* Demonstration. Suppose there are only two sets ; then 

 it is plain, that every quantity of one set, being combined with 

 every quantity of the other, will make all the compositions of 

 tv/o things, in these two sets ; and the number of these composi- 

 tions is evidently the product of the number of quantities in one 

 set by that in the other. 



Again, suppose there are three sets ; then the composition of 

 two, in any two of the sets, being combined with every quantity 

 of the third, will make all the compositions of 3 in the 3 sets. 

 That is, the compositions of 2, in any two of the sets, being mul- 

 tiplied by the number of quantities in tlie remaining set, will pro- 

 duce the compositions of 3 in the 3 sets ; which is evidently 

 the continual product of all the 3 numbers in the 3 sets. And 

 the same manner of reasoning will hold, let the number of sets be 

 what it will. (^E. I>. 



The doctrine of permutations, combinations, &c. is of very ex- 

 tensive use in different parts of the mathematics ; particularly in 

 the calculation of annuities and chances. The subject might have 

 been pursued to a much greater length ; but what has been done 

 already will be found sufficient for most of the purposes to which 

 things of this nature are applicable. 



