PERMUTATION AND COMBINATION. 233 



2. Take a series of as many terms, decreasing by i, 

 from the given number, out of which the election is to be 

 made, and find the product of all the terms. 



3. Divide the last product by the former, and the quo- 

 tient will be the number sought. 



EXAMPLES. 



In the same manner, it may be shewn, that the whole number 

 of combinations of 2, in 5 things, will be 14-24-3+4; of 2, 

 in 6 things, 1 4-24-3-^-44-5 ; and of 2, in 7, i 4-24-3+4+5 

 4-6, &c. 



Whence, universally, the number of combinations of m things, 

 taken 2 by 2, is =14-24-34-44-5-1-6, &c. to m — i terms. 



Bat the sum of this series is ::z!!L-^!!tII^ ' ; which is the same as 



I 2 



the rule. 



2. Let now the number of quantities In each combination be 

 supposed to be three. 



Then it is plain, that when mzz'^y or the things to be combined 

 are ahci there can be only one combination ; but if m be increased 

 by I, or the things to be combined be 4, as abcd^ then will the 

 number of combinations be increased by 3 ; since 3 is the num- 

 ber of combinations of 2 in all the preceding letters ahcj and with 

 each two of these the new letter d may be combined. 



The number of combinations, therefore, in this case, is 14-3. 



Again, if m be increased by one more, or the number of let- 

 ters be supposed 5 ;^hen the former number of combinatiojis will 

 be increased by 6 ; that is, by all the combinations of 2 in the 4 

 preceding letters, abed ; since, as before, with each two of these 

 the new letter e may be combined. 



The number of combinations, therefore, in this case, is 1 

 + 3 + 6. 



Whence, universally, the number of combinations of m things', 

 taken 3 by 3, is i + 3 4-6-f.io, 6cc. to m — 2 terms. 



But 



F F 



