I20 Algebra. 



;/ letters a, b, i\ d, <?, /, . . . taken ^ at a time by N,, we may 

 then write a before each of these N^ arrangements 5- at a time and 

 obtain N, arrangements ^+i at a time beginning with a. 



We may also write b before each of the same N, arrangements 

 and obtain N,, arrangements s-\- lat a time beginning with b, and 

 so on until each of the n letters a, b, c, d, . .is in turn placed be- 

 fore each of the N,- arrangements ^ at a time, and we then obtain 

 ?zN,- arrangements taken ^+i at a time, repetitions being allowed. 



Represent this number by N.,+ i and we have 



N,.^l=72N„ 



.y being a positive integer which may be greater or less than ;/. 



Giving s in turn all intermediate values from r— i down to i 

 and remembering that the number of arrangements one at a time 

 is equal to n, we have 



Nl = ?^. 

 Multiply these equals together and cancel the common factors 

 and we get 



8. Problem. To find the Number of Groups of 71 Things 

 Taken r at a Time, Repetitions being Allowed. 



To prepare the way for the general case we begin with the 

 groups of the four letters a, b, c, d taken three at a time, repeti-. 

 tions being allowed. 



In this case there are twenty groups, vis: 

 aaa aab aac aad abb 

 a be abd aee aed add 

 bbb bbe bbd bee bed 

 bdd eee eed edd ddd 

 Now if in eaeh of these twenty groups we leave the first letter 

 standing and advance the second letter one step and the third 

 letter two steps, we get twenty new groups of the six letters a, b, 

 e, d, e, /, as follows: 



