The sum $\sum _{\mathrm{i}=0}^{\mathrm{m}}\left(\begin{array}{c}10\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}20\\ \mathrm{m}-\mathrm{i}\end{array}\right)$ (where $\left(\begin{array}{l}\mathrm{p}\\ \mathrm{q}\end{array}\right)=0 \text{if }\mathrm{p}<\mathrm{q}$ ) is maximum when m is

$\sum _{\mathrm{i}=0}^{\mathrm{m}}\left(\begin{array}{c}10\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}20\\ \mathrm{m}-\mathrm{i}\end{array}\right)$

= Coefficient of x^m in (1 + x) ¹⁰ (1 + x) ²⁰ = ${ }^{30}{\mathrm{C}}_{\mathrm{m}}$

m = 15 (for maximum value)