The coefficient of $x^{7}$ in the expression

Question:

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

  1. (1) 210

  2. (2) 330

  3. (3) 120

  4. (4) 420


Correct Option: , 2

Solution:

The given series is in G.P. then

$S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$

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

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

$\therefore \quad$ Coefficient of $x^{7}$ is ${ }^{11} C_{7}={ }^{11} C_{11-7}={ }^{11} C_{4}=330$

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