The value of

Question:

The value of $\sum_{\mathrm{r}=0}^{6}\left({ }^{6} \mathrm{C}_{\mathrm{r}} \cdot{ }^{6} \mathrm{C}_{6-\mathrm{r}}\right)$ is equal to :

  1. 1124

  2. 1324

  3. 1024

  4. 924


Correct Option: , 4

Solution:

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

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

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

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

$=924$

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