Arrangements and Groups. 123 



Suppose now we have in all n letters, of which a is repeated r 

 times, b is repeated s times, c is repeated / times, and so on so 

 that r-\'S-\-t-{- . . . =71, and we wish to find the number of 

 arrangements taking all the 71 letters at a time. 



Fixing our attention upon any arrangement whatever of the 71 

 letters, let all the letters but the a's remain unchanged while the 

 r a's change places among themselves. Because all these a's are 

 alike we get only one arrangement, but if they had all been dif- 

 ferent we would have obtained \r arrangements, and since the 

 same thing is true whatever the arrangement upon which we 

 fixed our attention to begin with, it follows that there are | r 

 times as many arrangements when all the r letters are different as 

 there are under the present supposition. In the same way there 

 are | ^ times as many arrangements when the s b's are all different 

 as there are under the present supposition, also there are j t times 

 as many arr9.ngements when the t c's are all different as there are 

 under the present supposition, and so on. 



Hence there are \r- j ^ \t. . . times as many arrangements 

 when the 71 letters all are different as thtre are under the present 

 supposition, or the number of arrangements under t^e present 

 supposition is equal to the number of arrangements pf « things 

 taken all at a time, when all are different, divided by | r j ^ I / , . , 

 that is, the number of arrangements under the present supposition 

 is equal to 



\7t 



lllili- • • 



10, Probi^em. To fij^d the Number of Ways in which 

 « Things, no Two Alike, can be Made up into Sets of 

 which the first set contains r things, the second set contt^ius .y things, 

 the third contains / things, atid so on, where pf course 



r-^s + i-\- , . . =«. 



We begin with a special cas^ and find the nuipber pf w£^ys fiy^ 

 letters a, b, r, d, e, can be made up into twp sets of which the 

 first set contains two, and the second set three letters. 



Consider any particlar way of dividing into sets, say the first 

 set is ab, and the second set is cde. Then keeping the sets undis- 



