Question

The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$ is :

• A. $420$
• B. $330$
• C. $210$
• D. $120$

Answer 1

The correct option is B $330$

$(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$

Applying sum of terms of a G.P., we get

$(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$

$=\frac{(1+x{)}^{10}[1-{\left(\frac{x}{1+x}\right)}^{11}]}{(1-\frac{x}{1+x})}$

$=(1+x{)}^{11}-x^{11}$

$\therefore$ Coefficient of $x^7{\text{is}}^{11}{C}_7=330$