Question

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

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

Answer 1

The correct option is B

$330$

Step:1 Find out the relation between each terms

$a_1={(1+x)}^{10},a_2=x{(1+x)}^9,a_3==x^2{(1+x)}^8,....$

Therefore, all the terms of this series are in geometric progression with $a={(1+x)}^{10},r=\frac{x}{1+x},n=11$

Step:2 Calculating the sum of this G.P series

$\begin{array}{rcl}{S}_n& =& \frac{a[1-r^n]}{1-r}\\ & =& \frac{{(1+x)}^{10}\left[1-{\left(\frac{x}{1+x}\right)}^{11}\right]}{\left[1-\frac{x}{1+x}\right]}\\ & =& {(1+x)}^{11}-x^{11}\end{array}$

Step:3 Calculate the coefficient of $x^7$.

Therefore, the coefficient of $x^7$

$$\begin{gathered} =C_7^{11} \\ =\frac{11!}{7!\times 4!} \\ =\frac{11\times 10\times 9\times 8}{4\times 3\times 2} \\ =330 \end{gathered}$$

Therefore, Option(B) is the correct answer.