Arrangements and Groups. 115 



the letter c before each arrangement, and finally arrange the three 

 letters a, b, r in every possible way and place the letter d before 

 each arrangement. It is evident that all four letters a, b, c, d 

 appear in each arrangement thus formed, and it is also evident 

 that the number of arrangements in which a stands first is exactly 

 the same as the number in which b stands first, and so on. 



Hence there are in all four times as many arrangements of four 

 things taking all at a time as there are of three things taking all 

 at a time, or there are four times six or twenty-four arrangements 

 of four things taking all at a time. 



In geiieral, if we have n things, say the letters «, b,c, d, e,f,.. 

 then we may suppose all the letters but a arranged in every pos- 

 sible order and then a placed before each of these arrangements ; 

 then we may suppose all the letters but b arranged in every pos- 

 sible order and then b placed before each of these arrangements, 

 and so on. 



It is evident that all 71 letters appear in each arrangement thus 

 formed, and it is also evident that the number of arrangements in 

 which a stands first is exactly the same as the number in which 

 any other letter stands first. 



Now the number of arrangements in which a stands first is evi- 

 dently the number of arrangements of (n—\) things taken all at 

 a time, and hence the total number of arrangements of n things 

 taking all at a time is n times the number of arrangements of ;^— i 

 things taking all at a time. 



Let us represent the number of arrangements of 71 things taking 

 all at a time by A„ and the number of arrangements of 71— \ 

 things taken all a time by A„_i, etc. Then by what has just 

 been shown we have 



A,=;?A„_i, 



a„_i=,^«-i;a„_2, 

 A,_2=r«-2;A„_3, 



A3 = 3Ao, 

 A2 = 2A,, 

 A, = i. 



