Question

The coefficient of $x^{18}$ in the product $(1+x)(1-x{)}^{10}(1+x+x^2{)}^9$

• A. $84$
• B. $126$
• C. $-126$
• D. $-84$

Answer 1

The correct option is A $84$

$(1+x)(1-x{)}^{10}(1+x+x^2{)}^9=(1+x)(1-x){\left(\right(1-x\left)\right(1+x+x^2\left)\right)}^9$
$=(1-x^2)(1-x^3{)}^9$
Therefore the coefficient of $x^{18}$ in this expression will be
$=\text{coefficient of}{x}^{18}\text{in}(1-x^3{)}^9-\text{coefficient of}{x}^{16}\text{in}(1-x^3{)}^9$
Since, $(r+1{)}^{th}$ term in the expansion of
$(1-x^3{)}^9={}^9{C}_r(-x^3{)}^r={}^9{C}_r(-1{)}^r(x^{3r})$
Now for coefficient of $x^{18},3r=18\Rightarrow r=6$
and for coefficient of $x^{16},3r=16\Rightarrow r=\frac{16}{3}\notin N$
$\therefore$Required coefficient is ${}^9{C}_6=\frac{9!}{6!3!}=84$