£04 TROBLEM IN the DOCTRINE OF PERMUTATIONS, 



sort, several things of another sort, &c, by taking one at 

 a time, two at a time, &c, to any given number of things 

 at a time. 

 riL^^r"" I have not been able to find, that this problem has been 

 partially, considered by any authors, at least, that I am acquainted 



with, who have written on the doctrine of permutations and 

 combinations; except indeed Emerson, and one or two 

 other authors of a later date, who have a similar problem, 

 that is, a partial case of the above general one, which from 

 a repetition of operations would be sufficient for the solution 

 a*3 the rule' ^ f the present question, but the rule which is given by them 

 practice. for determining the number of combinations in each parti- 



'ctilar case is so long and tedious, that it is really of no use, 

 - beingiittle better, or less trouble, than finding the answer 

 from repeated trials. 

 Aver j- simple This circumstance led me to consider the problem inde- 

 jener ru e. p eRf ] Gn t| v f the measures there adopted, and having fallen 

 upon a very simple rule, which includes the particular case 

 of Emerson's in the general one above mentioned ; and as 

 it has not, to the best of my knowledge, been given by any 

 author, who has written on this subject, I have been in- 

 duced to submit it to you for insertion in your Journal, 

 should you think it deserving a place in that useful woxk. 



Problem. 



Problem. To determine the number of combinations, that can be 



formed out of a given number of things, in which there are 

 m things of one sort, n things of another sort, p things of 

 another sort, &c. ; by taking 1 at a time, 2 at a time, &c, 

 to any given number of things at a time. 



Rule. 



JLulz. Place in one horizontal row m + 1 units, annexing ci- 



phers on the right hand, till the whole number of units and 

 ciphers exceeds the greatest number of things to be taken 

 at a time by unity. 



Under each of these terms write the sum of the n -\- 1 

 left hand terms, including that as one of them, under which 

 the number is placed ; and under each of these write the 

 sum of the p 4- 1 loft hand terms of the last line. Under 



each 



