190 Summation of Polynomial Co -efficients, [May, 



(« 4. b + c + . . .) (d + &'+ c' 4. . . .) («" + 6" + c" + ...) = n 3 com- 

 binations of three letters each, by combining {a" + b" 4. c" + . . .) 

 with the preceding 



a" will form w 8 combinations of four letters each, 



b" will form 71" ditto of ditto, . • . then terms of the fourth fac- 

 tor will form n X n 3 ~ n* combinations of four letters each = S. of N. 

 C. of these combinations. 



Next, let there be five sets or factors, combining four at a time, and 

 taking four letters from each at a time, then 



5 4. 3. 2 



— - — ■ — - — z=. five arrangements of which the five sets are susceptible 



1. 2. 3. 4 



upon the condition required : but by the preceding result, each is 

 capable of n* combinations of four letters each ; the total number of 

 combinations ^ 5 re* ^= S. of N. C. 



m (m — \) (m — 2) (m — 3) 



Next combine m sets : then = number 



1.2 . 3.4 



m (m — 1) (m — 2) 



of arrangements : total number of combinations = 



1. 2. 3. 



(m— 3) 



- -m 4 =S. ofN. C. 



4 



The co-effic'ents of n* form the following series : 



Series 1 5 15 35 70 ? i!_^Zlll^-^_^Z±) 



1. 2. 3. 4 



4 10 20 35 m(m-l)(w-j) 



1. 2. 3 

 Differences^ 



I 6 10 15 21 *»(m-0 



L 1. 2 



4 5 6 m. 



Without going into any farther details, we may proceed to make 

 deductions : arranging the results already obtained, they will form the 

 following series : 



12 3 4 



■ m(m—l) , m(m— 1) (m— 2) , m(rn— \){m— 2)(m--3) » 



1. 2 1. 2. 3 1. 2. 3. 4 



rtli* 



m (m — 1) (m — 2) m — (r — 1) r 



1. 2. 3 ~ . . . r 

 The generating fraction is, for 



* rth means the rth term, as 2, 2, &c. mean the 1st, 2nd term, &c. 



