122 Algebra. 



These last formed groups are G("+''~^) in number, being formed 

 from that number of groups, and as the number of groups with 

 repetitions of n things taken r at a time has already been repre- 

 sented by N,-, hence G(""^"~^) cannot exceed n,.. 



It was previously proved that n^ could not exceed G("''~^), 

 hence, since neither can exceed the other, the number must be the 

 same, or, in other words, the number of groups of n things taken 

 r at a time, repetitions being allowed, is equal to the number of 

 groups of (7i-\-r—i) things taken r 2Lt a time without repetitions. 

 The last number has already been found. Hence the number of 

 groups of n things taking r at a time, repetitions being allowed, 

 equals 



(?i-\-r—i)(n-{-r—2) . . , n 



Vl 



which may be written in either of the forms 



n(n-\-\) . . . (7i-\-r—\) 



\n-\-r—\ 

 or 



n—\ r 



■Q. ProbIvKm. To find the Number of Arrangements 

 WHERE THE Given Things are not all Different. 



Illustration. — From what has gone before we know that the 

 number of arrangements of the letters a, b, c, d taken all at a time 

 is twenty-four, but if we have the letters a, a, b, c the number of 

 arrangements is only twelve. These twelve are the following : 

 aabc aacb abac abca 

 acab acba baac baca 

 bcaa caab caba cbaa 

 If we have the letters a, a, b, b there are only six arrange- 

 ments, viz : 



aabb abab abba 

 baab baba bbaa 

 If we have the letteas a, a, a, b there are only four arrange- 

 ments, viz : 



aaab aaba abaa baaa. 



Thus we see that with a given number of things the number of 

 arrangements depends upon how many of each kind are alike. 



