Determine the largest 3  digit prime factor of 2000 C 1000.
n C r denotes the number of combinations of n things taking r at a time.
The largest 3digit number is 661.
^{2000}C_{1000} = 2000!/1000!(20001000)! = 2000!/1000!(1000!)
=2000*1999*1998*....*1003*1002*1001/1000*999*998*....*3*2*1
=(1000*2)*1999*(999*2)*1997*....*(501*2)*1001/1000*....
= 2*1999*2*1997*..../500*499...
Leaving only odd numbers between 2000 and 1001 in the numerator and no number greater than 500 in the denominator. _ 2000/3 _ = 666. The largest prime between 500 and 666 is 661 (= 1983/3). Therefore 661 is the largest 3digit prime factor of ^{2000}C_{1000}.

Posted by Dej Mar
on 20060512 14:51:30 