The value of

$\sum_{r=0}^{6}{ }^{6} \mathrm{C}_{r} \cdot{ }^{6} \mathrm{C}_{6-r}$

$={ }^{6} \mathrm{C}_{0} \cdot{ }^{6} \mathrm{C}_{6}+{ }^{6} \mathrm{C}_{1} \cdot{ }^{6} \mathrm{C}_{5}+\ldots \ldots+{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{0}$

Now,

$(1+x)^{6}(1+x)^{6}$

$=\left({ }^{6} \mathrm{C}_{0}+{ }^{6} \mathrm{C}_{1} \mathrm{x}+{ }^{6} \mathrm{C}_{2} \mathrm{x}^{2}+\ldots \ldots+{ }^{6} \mathrm{C}_{6} \mathrm{x}^{6}\right)$

$\left({ }^{6} \mathrm{C}_{0}+{ }^{6} \mathrm{C}_{1} \mathrm{x}+{ }^{6} \mathrm{C}_{2} \mathrm{x}^{2}+\ldots \ldots+{ }^{6} \mathrm{C}_{6} \mathrm{x}^{6}\right)$

Comparing coefficeint of $x^{6}$ both sides

${ }^{6} \mathrm{C}_{0} \cdot{ }^{6} \mathrm{C}_{6}+{ }^{6} \mathrm{C}_{1}+{ }^{6} \mathrm{C}_{5}+\ldots \ldots . .+{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{0}={ }^{12} \mathrm{C}_{6}$

$=924$