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) 210
B) 420
C) 120
D) 330

Answer 1

Here we need to use both geometric progression and combination because we have to find the coefficient of that term. A term is made of a variable with the coefficient of that variable.

Complete step by step solution:
Given the series,
${(1 + x)^{10}} + x{(1 + x)^9} + {x^2}{(1 + x)^8}....... + {x^{10}}$
Here common ratio r= $\dfrac{{x{{(1 + x)}^9}}}{{{{(1 + x)}^{10}}}}$
r$\Rightarrow \dfrac{x}{{1 + x}}$
First term, $a={(1 + x)^{10}}$
Here there are 11 total terms.
Now sum of all these terms in G.P.is given by
S=$a\dfrac{{1 - {r^n}}}{{1 - r}}$
$S = {(1 + x)^{10}}\left( {\dfrac{{1 - {{\left( {\dfrac{x}{{1 + x}}} \right)}^{11}}}}{{1 - \dfrac{x}{{1 + x}}}}} \right)$

$$\Rightarrow S = {(1 + x)^{10}}\left( {\dfrac{{\dfrac{{{{\left( {1 + x} \right)}^{11}} - {x^{11}}}}{{{{\left( {1 + x} \right)}^{11}}}}}}{{\dfrac{{1 + x - x}}{{1 + x}}}}} \right) \\\ \Rightarrow S = {(1 + x)^{10}}\left( {\dfrac{{{{\left( {1 + x} \right)}^{11}} - {x^{11}}}}{{{{(1 + x)}^{10}}}}} \right) \\\ \Rightarrow S = {\left( {1 + x} \right)^{11}} - {x^{11}} \\\$$

Now coefficient of ${x^7}$ is given by

$$11{C_7} = \dfrac{{11!}}{{7!\left( {11 - 7} \right)!}} \\\ \Rightarrow \dfrac{{11!}}{{7!\left( 4 \right)!}} \\\ \Rightarrow \dfrac{{11 \times 10 \times 9 \times 8}}{{24}} \\\ \Rightarrow 330 \\\$$

So the coefficient of ${x^7}$ is 330.

So option D is correct.

Note:

1. In a geometric progression except for the first term, other terms are obtained by multiplying the previous term with a fixed common ratio.
2. $a,ar,a{r^2},.....$ is a geometric progression.
3. This common ratio is denoted by r and the first term is denoted by a.
4. If three positive numbers a, b, c are in G.P. then their geometric mean is b. such that $b= \sqrt {ac}$.
5. An arithmetic progression is having a common difference d.
   $a,a + d,a + 2d....$ is an arithmetic progression.