Mia Pitts

2023-02-26

The coefficient of ${x}^{7}$ in the expression $\left(1+x{\right)}^{10}+x\left(1+x{\right)}^{9}+{x}^{2}\left(1+x{\right)}^{8}+...+{x}^{10}$ is :
A)420
B)330
C)210
D)120

Darien Jennings

The right decision is B 330
$\left(1+x{\right)}^{10}+x\left(1+x{\right)}^{9}+{x}^{2}\left(1+x{\right)}^{8}+...+{x}^{10}$
Applying sum of terms of a G.P., we obtain
$\left(1+x{\right)}^{10}+x\left(1+x{\right)}^{9}+{x}^{2}\left(1+x{\right)}^{8}+...+{x}^{10}$
$=\frac{\left(1+x{\right)}^{10}\left[1-\left(\frac{x}{1+x}{\right)}^{11}\right]}{\left(1-\frac{x}{1+x}\right)}$
$=\left(1+x{\right)}^{11}-{x}^{11}$
$\therefore$ Coefficient of ${x}^{7}$ is ${}^{11}{C}_{7}=330$

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