n C r denotes the number of combinations of n things taking r at a time.

2000 C 1000 is 2000*1999*...*1001 / 1000!, we are looking for prime factors that are prime factors of more of the 1000 possibly prime defining factors of the numerator than of the 1000 possibly prime defining factors of the denominator.

Let's assume the prime factor answer is larger than 500, and so is between 500 and 999 (we could go back later if this were not true).  Such a prime factor would occur only once within 1000!, as twice such a number would be larger than 1000.

One times such a number would be too small for the numerator, but twice such a number would be found there. So far that just cancels the one time it occurs in the denominator.  But if the number is small enough, three times it will also be found as one of the defining factors in the numerator. It only needs to be less than 2/3 of 1000. The largest such is 661. It occurs as a factor twice in the numerator but only once in the denominator.

Edited on May 13, 2006, 12:39 pm

Posted by Charlie on 2006-05-13 12:38:15