Question

The value of $\left({}^7{C}_0+{}^7{C}_1\right)+\left({}^7{C}_1+{}^7{C}_2\right)+\cdots +\left({}^7{C}_6+{}^7{C}_7\right)$ is

• A. $2^8-2$
• B. $2^8-1$
• C. $2^8+1$
• D. $2^8$

Answer 1

Answer: (A)

$\left({}^7{C}_0+{}^7{C}_1\right)+\left({}^7{C}_1+{}^7{C}_2\right)+\cdots +\left({}^7{C}_6+{}^7{C}_7\right)$
$={}^8{C}_1+{}^8{C}_2+\cdots +{}^8{C}_7={}^8{C}_0+{}^8{C}_1+{}^8{C}_2+\cdots +{}^8{C}_7+{}^8{C}_8$
$-\left({}^8{C}_0+{}^8{C}_8\right)=2^8-1(1+1)=2^8-2$