126 Algebra. 



and from this it follows that 



A("-.)- r„j|+7 ■ ^4; 



But Iw— r+i equals the product of the integer numbers from 

 I up to ?i—r-i-i, and this product of course equals 2t—r-\-i 

 times the product of the integer numbers from i up to «— r, or 



]n-^r-^i = (?i—r+i) \n-r-r, 



\n 

 lience A(;.'_i)=-7 }^=^—, (s) 



Comparing this with the value of A("), equation (^3^, we get 



Ac;)=r«-r+i;AC'_,). (6) 



If in (■^) we make r=?i we get 



A(;;)=A(;;_,), (-j) 



or the number of arrangements of 71 things taken all at a time 

 equals the number of arrangements of 71 things all but one at a 

 time. 



We have already found that 



\n 

 \r \7i — r 



and from this it follows that 



I n-i 



^^(r')=v:~==--- (9) 



n^r—i 



Multiply both numerator and denominator of this last fraction 



by 71(71 — r), remembering that 7i i ;? — i = 1 71 and that (7i—r)\ n—r—\ 

 ^ |^?-^r, ^nd we get 



n\r \n—r , 



hence, from (S) and (g), 



7t-^ r 

 From (8 j it easily follows that 



ar" "1= t=— ■ ( 12) 



