The value of

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

Now,

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

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

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

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

${ }^{6} \mathrm{C}_{0} \cdot{ }^{6} \mathrm{C}_{6}+{ }^{6} \mathrm{C}_{1}+{ }^{6} \mathrm{C}_{5}+\ldots \ldots+{ }^{6} \mathrm{C}_{6}{ }^{66} \mathrm{C}_{0}={ }^{12} \mathrm{C}_{6}$

$=924$