Question

The value of $\left({}^{7}C_0+{}^{7}C_1\right)+\left({}^{7}C_1+{}^{7}C_2\right)+...+\left({}^{7}C_6+{}^{7}C_7\right)$ is
(a) 2⁷ − 1
(b) 2⁸ − 2
(c) 2⁸ − 1
(d) 2⁸

Answer 1

(b) 2⁸ − 2


$\left({}^{7}C_0+{}^{7}C_1\right)+\left({}^{7}C_1+{}^{7}C_2\right)+\left({}^{7}C_2+{}^{7}C_3\right)+\left({}^{7}C_3+{}^{7}C_4\right)+\left({}^{7}C_4+{}^{7}C_5\right)+\left({}^{7}C_5+{}^{7}C_6\right)+\left({}^{7}C_6+{}^{7}C_7\right)$

$=1+2\times {}^{7}C_1+2\times {}^{7}C_2+2\times {}^{7}C_3+2\times {}^{7}C_4+2\times {}^{7}C_5+2\times {}^{7}C_6+1$

$=1+2\times {}^{7}C_1+2\times {}^{7}C_2+2\times {}^{7}C_3+2\times {}^{7}C_3+2\times {}^{7}C_2+2\times {}^{7}C_6+1$

$$\begin{gathered} =2+2^2\left({}^{7}C_1+{}^{7}C_2+{}^{7}C_3\right) \\ =2+2^2\left(7+\frac{7}{2}\times 6+\frac{7}{3}\times \frac{6}{2}\times 5\right) \end{gathered}$$

$$\begin{gathered} =2+252 \\ =254 \\ =2^8-2 \end{gathered}$$