Question

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

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

Answer 1

Answer: (D)

$\sum _{x=0}^6{}^6{C}_r\cdot {}^6{C}_{6-r}$
$={}^6{C}_0\cdot {}^6{C}_6+{}^6{C}_1\cdot {}^6{C}_5+\dots \dots +{}^6{C}_6\cdot {}^6{C}_0$
Now,
$(1+x{)}^6(1+x{)}^6$
$=\left({}^6{C}_0+{}^6{C}_{1}x+{}^6{C}_2{x}^2+\dots ..+{}^6{C}_6{x}^6\right)$
$\left({}^6{C}_0+{}^6{C}_{1}x+{}^6{C}_2{x}^2+\dots \dots +{}^6{C}_6{x}^6\right)$
Comparing coefficeint of $x^6$ both sides
${}^6{C}_0\cdot {}^6{C}_6+{}^6{C}_1+{}^6{C}_5+\dots \dots .+{}^6{C}_6\cdot {}^6{C}_0={}^{12}{C}_6$
$=924$