Các số hạng có dạng:
$(k+1)C^k_{2000}= $$\frac{(k).2000!}{k!.(2000-k)!}+C^k_{2000}=$ $\frac{2001!}{2001.(k+1)!.(2000-k)!}+C^k_{2000}=$ $\frac{1}{2001}.C^{k+1}_2001+C^k_{2000}$
Suy ra: $S=\frac{1}{2001}. (C_{2001}^1+C_{2001}^2+...+C_{2001}^{2001})+(C^0_{2000}+C^1_{2000}+...+C^{2000}_{2000})$
$=\frac{1}{2001}.(2^{2001}-C_{2001}^0)+2^{2000}$
$=\frac{2003.2^{2000}-1}{2001}$