The value of r for which

Consider the expression ${ }^{20} C_{r}{ }^{20} C_{0}+{ }^{20} C_{r}-1{ }^{20} C_{1}$ $+{ }^{20} C_{r-2}{ }^{20} C_{2}+\ldots+{ }^{20} C_{0} \cdot{ }^{20} C_{r}$

For maximum value of above expression $r$ should be equal to 20 .

as ${ }^{20} C_{20} \cdot{ }^{20} C_{0}+{ }^{20} C_{19} \cdot{ }^{20} C_{1}+\ldots+{ }^{20} C_{20} \cdot{ }^{20} C_{0}$

$=\left({ }^{20} C_{0}\right)^{2}+\left({ }^{20} C_{1}\right)^{2}+\cdots+\left({ }^{20} C_{20}\right)^{2}={ }^{40} C_{20}$

Which is the maximum value of the expression,

So, $r=20$