1832.] 



Summation of Polynomial Co-ejjicients. 



191 



Term 

 2.. 



3 



m — 1 



~ir 



m — 2 

 ' 3 

 m — 3 



m — (r — 1) 



m- 



{n+i> 



Let N be the number of terms ; 

 the generating fraclion of ihe 



N -f ] th term 



N 



Since the ]\ -f-itii term = o, the ge- 

 nerating fraction must =: o y 

 Therefore 



-] 



That is, the number of terms is equal to the number of factors, or 

 sets employed or developed. 



The index of n therefore in the last term is m; i. e. in tlie last term 

 n is n m ; as for the co-efficient of w m , it is equal to the sum of the com- 

 binations that can be formed by combining m sets or factors, taking N 

 sets at a time. 



m (m—l) m— (N — 1) __ 



This sum is equal to 



N 



1.2. 3 



(because N = rri) 



m (m — 1) 1. 2. 3 __ - 



1. 2. 3 ~Jm — 1) m 



therefore the last term is nm r= (S. of N. Co-efficients of all its combi- 

 nations. As the first term has n combinations and the last n m , the last 

 but one, or the N — lth term will have p- 1 , and its co-efficient will 

 be equal to the sum of the combinations of m sets or factors taken N — 1 



sets at a time: 



jn r m 



1. 



m (m — 1) (m- 

 1. 2. 3. 



__1) ™ — \(m — 1) — 11 



""" J™=\T~ 



-2 

 4 



(w— 2(m— 1) 

 fore the N — lth term^= m n™—> = S. N. C. 



-I 



1)3 



= m = S. N. C. there- 



_7«(m 1) m s 



N— 2 .... ~ m \ m — l ) n m — * &c. &c. 

 = S.N.C. 1. 2 



Since the foregoing formulae for the number of combinations also 



represent the sum of the numeral co-efficients of each combination, these 



formula?, considered as representing the sum of the combinations, will 



not at all be affected by the hypothesis that the n terms of each of 



the m sets or factors is the same, viz. that «', a" &c. — a ; 6', b" &c. 



= b ; d, c" = c and so on : the formula? will, under this supposition, 



continue to represent the sum of the numeral co-efficients. 



