Question

The co-efficient of $x^4$ in the expansion of ${\left(1-x+2x^2\right)}^{12}$ is :

• A. ${}^{12}{C}_3$
• B. ${}^{13}{C}_3$
• C. ${}^{14}{C}_4$
• D. ${}^{12}{C}_3+3{}^{13}{C}_3+{}^{14}{C}_4$

Answer 1

Answer: (D)

${\left(1-x+2x^2\right)}^{12}={\left((1-x)+2x^2\right)}^{12}$
$=(1-x{)}^{12}+{}^{12}{C}_1(1-x{)}^{11}2x^2+{}^{12}{C}_2(1-x{)}^{10}4x^4+\dots \dots$
$\text{ required co-efficient of }{x}^4=$
$\text{ co-eff. of }{x}^4\text{ in }(1-x{)}^{12}+24\text{ co-eff. of }{x}^2\text{ in }(1-x{)}^{11}+4\cdot {}^{12}{C}_2$
$={}^{12}{C}_4+24\cdot {}^{11}{C}_2+4\cdot {}^{12}{C}_2={}^{12}{C}_4+\frac{2\cdot 12\cdot 11\cdot 10\cdot 3}{1\cdot 2\cdot 3}+4\cdot {}^{12}{C}_2$
$={}^{12}{C}_4+6\cdot {}^{12}{C}_3+4\cdot {}^{12}{C}_2={}^{12}{C}_4+2\cdot {}^{12}{C}_3+4\cdot {}^{12}{C}_3+{}^{12}{C}_2)$
$={}^{12}{C}_4+2\cdot {}^{12}{C}_3+4\cdot {}^{13}{C}_3={}^{12}{C}_4+{}^{12}{C}_3+{}^{12}{C}_3+4\cdot {}^{13}{C}_3$
$={}^{13}{C}_4+{}^{13}{C}_3+{}^{12}{C}_3+3\cdot {}^{13}{C}_3={}^{14}{C}_4+3\cdot {}^{13}{C}_3+{}^{12}{C}_3\Rightarrow D$