Arrangements and Groups. 12 j 



abc abd abe abf acd 

 ace acf ade adf aef 

 bed bee bef bde bdf 

 bef ede edf eef def 



The groups here written are the groups of the six letters a, b, 

 c^ d, e, /, without repetitions. 



In a similar manner we may deal with the general case of the 

 number of groups of n letters a, b, c, d, e,f, . . . taken r at a 

 time, repetitions being allowed. I^et the number of these groups 

 be denoted by n^ and suppose them all written down in alpha- 

 betical order ; then in eaek of these groups keep the first letter un- 

 changed, advance the second letter one step, the third letter two 

 steps, the fourth letter three steps and so on. 



We thus form n^ new groups containing all the letters the orig- 

 inal ones contained, and r— i other letters. These new groups 

 are written in alphabetical order, because the original ones were, 

 and by the way in which the letters have been advanced it is evi- 

 dent that no letter is repeated in any one of these new groups. 



No two of these new groups are alike, else two of the original 

 groups would have been alike. 



Now since each of these new groups contain r of the n-\-r — i 

 letters a, b, c, d, e, , . . , and since no letter is repeated in any 

 group, and since no two groups are alike, therefore these new 

 groups constitute some or all of the groups of the Jt 4- r— i letters 

 a, b, e, d, e, . . . taken r at a time without repetitions. 



Let the number of groups without repetitions of n-\- r— i things 

 taken rat a time be represented by G("+''~^j then it is evident that 

 N;. cannot exceed G("^''~^). Now let us conceive each of the 

 G("^''~0 groups written down in alphabetical order, and then 

 leave the first letter in each group unchanged, change the second 

 letter in each group to the one just before it in the alphabet, the 

 third one in each group to the second one before it in the alpha- 

 bet and so on, then these groups are changed into new groups 

 wherein some of the letters are repeated, but no letter is beyond 

 the wth letter of the alphabet. Moreover no two of these groups 

 are alike, ka nc a no two of those from which they were formed *^«'^^'^^ 

 were alike, so that these new groups must be some or all of the^^ ^^'^'^^ 

 n letters a, b, c, d, e, . . . taken r at a time with repetitions. 



A— 15 



