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

The largest 3-digit number is 661.

²⁰⁰⁰C₁₀₀₀ = 2000!/1000!(2000-1000)! = 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 3-digit prime factor of ²⁰⁰⁰C₁₀₀₀.