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

Hint:
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.