^3^ ARITHMETIC. 



PROBLEM V, 



fn* find the mfmher of cGmbinatioin of any given nuinler of 

 thitigSy all different from one a?ioiher^ tahen any given num' 

 her at -a time. 



I. Take the series ij 2, 3, 4, &c. up to the number to 

 be taken at a time, and find the product of all the terms. 



2. Take 



* This rule, expressed algebraically, is — X X X 



^Lni. , Sec. to s^ .terms 1 where m is the number of given quanti- 



4 

 tics, and n those to be takeij at a time. . 



Demonstkation of the Rule, i . Let the number of things 

 to be taken at a time be 2, and the things to be combined =;n. 



Now, when m, or the number of things to be combined, is on- 

 ly two, as a and by it is evident, that there can be only one com- 

 bination, as ab ; but if m be increased by i, or the letters to be 

 combined be 3, as abc, then it is plain, that the number of com- 

 binations will be increased by 2, since with each of the former 

 letters, a and ^, the new letter c may be joined. It is evident, 

 therefore, that the whole number of combinations, in this case, 

 will be truly expressed by i -|- 2. 



Again, if m be increased by one letter more, or the whole 

 number of letters be four, as abed ', then it will appear, that the 

 whole number of combinations must be increased by 3, since 

 with each of the preceding letters the new letter d may be com- 

 bined. The combinations! therefore, In this case, will be truly 

 expressed by i -f- 2 -|- 3. 



