Question

The value of $({}^7{C}_0+{}^7{C}_1)+({}^7{C}_1+{}^7{C}_2)+...+({}^7{C}_6+{}^7{C}_7)$

• A. $2^7-1$
• B. $2^8-2$
• C. $2^8-1$
• D. $2^8$

Answer 1

The correct option is B

$2^8-2$

$({}^7{C}_0+{}^7{C}_1)+({}^7{C}_1+{}^7{C}_2)+({}^7{C}_2+{}^7{C}_3)+({}^7{C}_3+{}^7{C}_4)+({}^7{C}_4+{}^7{C}_5)+({}^7{C}_5+{}^7{C}_6)+({}^7{C}_6+{}^7{C}_7)$

$=1+2\times {}^7{C}_1+2\times {}^7{C}_2\times {}^7{C}_3+2\times {}^7{C}_4+2\times {}^7{C}_5+2\times {}^7{C}_6+1$

$=1+2\times {}^7{C}_1+2\times {}^7{C}_2\times {}^7{C}_3+2\times {}^7{C}_3+2\times {}^7{C}_2+2\times {}^7{C}_6+1$

$=2+2^2({}^7{C}_1+{}^7{C}_2+{}^7{C}_3)$

$=2+2^2(7+\frac{7}{2}\times 6+\frac{7}{3}\times \frac{6}{2}\times 5)$

$=2+252=254$

$=2^8-2$