1 92 Summation of Polynomial Co-efficients. [Mat, 



Now let it be required to dev elope m sets or factors, each consist- 

 ing of n things or terms ; viz. 



(x -f a -f b -f. c &c.) (x + a' -f b' + C -f &c.) (x .|_ a" -f 6" + &c.) 

 (&c.) it is plain, that the developement will be 



X m + A^ m -' +B*™- 2 + Cx m - 3 + D* »-* 



Y x* -\-Y x -\- Z. Here it is evident, that, 



A = (a -j- b -f- &c. a' -f- 6' + &c. + a" + 6" + &c.) = the sum of 

 the combinations of m sets of n letters each, one set taken at a time, 

 and one letter from that set at a time ; = (as is shewn above) the sum 

 of m n combinations of one letter each. 



B = (aa' + aJ' + &c. . . aa" + ab" + &c. . . a' a" + a' b" &c.) 

 === S. of the combinations of m sets of n letters each, two sets taken at 

 a time, and one letter from each of these two sets taken at a time ; = 



S. oflSl l Ill} n* combinations of two letters each. 

 1. 2. 

 C = (aa' a" + a b'b" + &c. b' a' a» + a' a" a'" . . &c.) = S. of 



the combinations of m sets of n letters each ; three sets taken at a time, 



and one letter from each of these three sets combined at a time) = S. 



n f \ m ) y m "i n 3 combinations of three letters each. 

 1. 2. 3 



D = S. of __A ZT—2.A JLl 1 n x combinations of four letters 



1. 2. 3. 4 



each. 



Z = S. of the combinations of m sets of n letters each, taken m sets 

 at a time, and one letter from each at a time =1 X n m =z n m combina- 

 tions of m letters each. 



Y = S. of the combinations of m sets of n letters ; taken (m — 1 ) sets 

 at a time, and one letter at a time = m n m — 1 combinations of (m — 1) 

 letters each, &c. 



Now, if we suppose a = a' = a" &c. b = b' =. b" &c. c = c' = c" 

 &c. m n will represent the sum of the numeral co-efficients in A ; 



— 5: 1 n? the sum of the numeral co-efficients in B ; therefore 



the sum of the numeral co-efficients is in 



X 1 



A a; m — • mn 



Bl^ m t™— ^ ni 



1. 2 



C^-» m (m—1) (w— 2 ) n% 



1. 2. 3 



