Question

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

• A. 1124
• B. 1324
• C. 1024
• D. 924

Answer 1

Answer: (D)

$\displaystyle\sum_{x=0}^{6}{ }^{6} C _{ r } \cdot{ }^{6} C _{6- r } $
$=\,{ }^{6} C _{0} \cdot { }^{6} C _{6}+{ }^{6} C _{1} \cdot{ }^{6} C _{5}+\ldots \ldots+{ }^{6} C _{6} \cdot{ }^{6} C _{0}$
Now,
$(1+x)^{6}(1+x)^{6} $
$=\left({ }^{6} C_{0}+{ }^{6} C_{1} x+{ }^{6} C_{2} x^{2}+\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 $x^{6}$ both sides
${ }^{6} C _{0} \cdot{ }^{6} C _{6}+{ }^{6} C _{1}+{ }^{6} C _{5}+\ldots \ldots .+{ }^{6} C _{6} \cdot{ }^{6} C _{0}={ }^{12} C _{6} $
$=924$