433 



Some Properties of Binomial Coefficients. 



A. M. Kenyon. 



The binominal coefficients of the expansion 



, , a- (k) k . (k) ft— l . fft] k—2 2 . . (ftl k 



(x + y) = [oj x + [i} x y+ {2\ x y + ■ ■ ■ + [k\y 



were known to possess a simple recursion formula 



(1) 



+ 



fc + ll 



In + lj ' [n + lj 



k, n = 0, 1, 2, 3 



by means of which Pascal's Triangle* 





n = 



n = 1 



n = 2 



« = 3 . 



n = 4 



etc. 



ifc = 



1 













k = 1 



1 



1 











k = 2 



1 



2 



1 









k = 3 



1 



3 



3 



1 







ft = 4 



1 



4 



6 



4 



1 





etc. 



— 



— 



— 



! 



— 



— 



could be built up, before Newton showed that they are functions of ft and n: 



ft] ft(ft-l) . . . (k-n + 1) 



ft = 0, 1 : 2 



(2 





1 



-, »-l, 2, 3, 

 re = 



A great number of relations involving binomial coefficients have been 

 discovered**; some of the most useful of these are 



(3) nf*l-* r *-J ; k = k -\ k - V \; [*l = 0if*>ft. 



•See Chrystal : Algebra I, p. 81. 



**See Chrystal : Algebra IT, Chaps. XXIII, XXVII. Hagen : Synopsis der hoeheren 

 Mathematik, p. 64; Pascal: Repertorium der hoeheren Mathematik I, Kap. II, Sec. I. 



28—4966 



