The coefficient of x⁷ in the expression $(1+\mathrm{x}{)}^{10}+\mathrm{x}(1+\mathrm{x}{)}^9+{\mathrm{x}}^2(1+\mathrm{x}{)}^8+\dots +{\mathrm{x}}^{10}$ is

detailed solution

Correct option is B

Given expression (1+x)10+x(1+x)9+x2(1+x)8+…+x10The above expression is geometric series having first term,a=(1+x)10Common ratio,r=x1+x and number of terms, n = 11sum of geometric series =a1−rn1−r=(1+x)101−x1+x111−x1+x                                         =(1+x)10(1+x)11−x11(1+x)111+x−x1+x=(1+x)11−x11The coefficient of x7 an the given expression= coefficient of x7 in the expansion of (1+x)11−x11= coefficient of x7 in the expansion of (1+x)11=11C7=11×10×9×84×3×2=11×10×3=330