Arrangements and Groups. 117 



Then every arrangement in this class contains besides <2, (r—i) 

 of the letters b, c, d, . . , and since a is fixed while the remain- 

 ing letters are arranged in every possible order, therefore the 7iitmber 

 of arrangements in the class considered must equal the number of 

 arrangements of 71 — \ letters b, c, d, . . , taken r— i at a time. 



As there are 71 such classes and as the number of arrangements 

 in each class equals the number of arrangements of 71—1 things 

 taking r— i at a time, therefore the total number of arrangements of 

 n things taken r at a time equals 7i times the number of arrange- 

 ments of ?^— I things taken r—i at a time. 



Let us represent the number of arrangements of ;/ things taken 

 r at a time by A(") and similarily any number of things taken 

 anj^ number at a time, say ^ things taken / at a time (s being 

 greater than O t>y A(;), then by what has just been proved 



A0) = 7iA0z\) 

 A(-}) = (7i-i)A('r:l) 



ACr'-') = (n-7^-\-i). 



Multipl}^ these equations together, member by member, and 



cancel common factors and we get 



A(i':.) = 7i(7i—j)(7i — 2) . . . (n — r-\-i). 



Multiply and then divide the right-hand member by 



(7t — r)(ti — r-{-i) ... I and we get 



_7i(n—i)(7i — 2) . . . (n — i'-\-i)(7i — r)(n — r—\). . . i 



A(") — — — -. -; - 



[71 — r)(7i — r — I ... I 



It is easily seen that the numerator is 1 71 and the denominator 

 is \n—r, hence 



\n 



A{;o=,--^ 



71 — 7' 



4. Problem. To find the Number of Groups of ;/ Dif- 

 ferent Things taken r at a Time. 



. Take the letters a, b, c, d, e, . . . , and suppose the groups all 

 written down ; then, fixing our attention upon any one group, it is 

 evident that there could be several different arrangements made 

 from that group by changing the order of the letters. 



