If   $\pi \left(n\right)$ denotes the product of all binomial coefficients in ${\left(1+x\right)}^n$ , then the ratio of $\pi \left(2002\right)$   to  $\pi \left(2001\right)$ is

$\frac{\pi \left(2002\right)}{\pi \left(2001\right)}=\frac{{ }^{2002}{c_0}^{.2002}{c_1}^{.2002}{c}_2{\mathrm{....}}^{2002}{c}_{2002}}{{ }^{2001}{c_0}^{.2001}{c_1}^{.2001}{c_2}^{\mathrm{......2001}}{c}_{2001}}$

$=\frac{{ }^{2002}{c}_0}{{ }^{2002}{c}_0}.\frac{{ }^{2002}{c}_1}{{ }^{2001}{c}_1}.\frac{{ }^{2003}{c}_2}{{ }^{2001}{c}_2}\mathrm{......}\frac{{ }^{2002}{c}_{2001}}{{ }^{2002}{c}_{2001}}{.}^{2002}{c}_{2002}$

$=\frac{2002}{2002}.\frac{2002}{2001}.\frac{2002}{2000}.\frac{2002}{1999}\mathrm{.....}\frac{2002}{1}.1=\frac{{\left(2002\right)}^{2001}}{\left(2001\right)!}$