The value of $\sum_{r=0}^{6}\left({ }^{6} \mathrm{C}_{r}{ }^{-6} \mathrm{C}_{6-\mathrm{r}}\right)$ is equal to :
Correct Option: , 4
$\sum_{r=0}^{6}{ }^{6} \mathrm{C}_{\mathrm{r}} \cdot{ }^{6} \mathrm{C}_{6-\mathrm{r}}$
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
${ }^{6} \mathrm{C}_{0} \cdot{ }^{6} \mathrm{C}_{6}+{ }^{6} \mathrm{C}_{1}+{ }^{6} \mathrm{C}_{5}+\ldots \ldots+{ }^{6} \mathrm{C}_{6}{ }^{66} \mathrm{C}_{0}={ }^{12} \mathrm{C}_{6}$
$=924$
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