Question

Let $a$ and $b$ be the coefficient of $x^3$ in ${\left(1+x+2x^2+3x^3\right)}^3$ and ${\left(1+x+2x^2+3x^3+4x^4\right)}^3$, respectively then

• A. $a=b$
• B. $a>b$
• C. a $<$ b
• D. $a+b=44$

Answer 1

Answer: (A), (D)

$(1+z{)}^3\text{ where }z=x\left(1+2x+3x^2\right)$
$1+{}^3{C}_{1}z+{}^3{C}_2{z}^2+{}^3{C}_3{z}^3$
$\text{ coefficient of }{x}^3\text{ in }(1+z{)}^3$
${}^3{C}_1(3)+{}^3{C}_2(4)+{}^3{C}_3(1)=22$
$\Rightarrow a=22$
$\text{ now again }(1+y{)}^3$
$\text{ where }y=x\left(1+2x+3x^2+4x^3\right)$
$(1+y{)}^3=1+{}^3{C}_{1}y+{}^3{C}_2{y}^2+{}^3{C}_3{y}^3$
$\therefore \text{ coefficient of }{x}^3\text{ is }$
${}^3{C}_1(3)+{}^3{C}_2(4)+{}^3{C}_3(1)$
$=9+12+1=22$
$\Rightarrow b=22$
$\text{ Hence a }=b\Rightarrow a+b=44$