WARixa 451 



suppose that a, h, c, d, ... are the chances that the numbers 

 ®> A 7j ^> • • • respectively will occur in one trial ; and then N is the 

 chance that in n trials the sum of the numbers will be tt. 



835. "Waring gives on his page 22 the theorem which we 

 now sometimes call by the name of Yanclermonde. The theorem 

 is that 



(a + Z>) (a + Z»-l) ... (a + Z'-w+ 1) 

 = a (a — 1) .,. [a — n+l) 

 -f wa (a — 1) . .. (a — n 4- 2) J 



+ 



+ ^^(^-1)... {l-n+l). 



From this he deduces a corollary which we will give in our 

 own notation. Let <f> {x, y) denote the sum of the products that 

 can be made from the numbers 1. 2, 3, ... x, taken y together. 

 Then will 



Li 



S — 1 



- (f) ill — 1, ?l — 5) 



= ; — • <h (n — r — \, n — s) 



I r \n — r ^ ^ 



+ ^, ^ -<l>{n-r-%n-s-\)<i>[r,l) 



r+1 n — r — L 



+ ..,^ - ^^(n-r-3,?i-5-2)0(r+l, 2) 



r + 2 [ 



-r-2 



n — r 



+ — TTTT^ ^i>{n-r-i,n-s-9)4>{r+%'i) 



+ 



It must be observed that 5 is to be less than n, and r less than 

 s ; and the terms on the right-hand side are to continue until we 

 arrive at a term of the form ^ {x, 0), and this must be replaced 

 by unity. 



29—2 



