If the coefficient of x in the expansion of (ax^2+bx+c)(1-2x)^26 is -56 and the coefficien…

Question

If the coefficient of $x$ in the expansion of $(a{x}^{2}+bx+c)(1-2x{)}^{26}$ is -56 and the coefficients of ${x}^{2}$ and ${x}^{3}$ are both zero, then $a+b+c$ is equal to :

Accepted Answer

$\begin{gathered} (a{x}^{2}+bx+c)(1-2x{)}^{26} \ a{x}^{2}(1-2x{)}^{26}+bx(1-2x{)}^{26}+c(1-2x{)}^{26} \ \ x\Rightarrow [0+b\cdot {}^{26}{C}_{0}+c\cdot {}^{26}{C}_{1}(-2)]x \ =b-52c $

Coefficient of

$\begin{gathered} {x}^{2}=(a({}^{26}{C}_{0})+b\cdot {}^{26}{C}_{1}(-2)+c{}^{26}{C}_{2}(-2{)}^{2}){x}^{2} \ =a-52b+{}^{26}{C}_{2}	imes 4=a-52b+1300c \end{gathered}$

Coefficient of

$ {x}^{3}=[a\cdot {}^{26}{C}_{1}(-2)+b\cdot {}^{26}{C}_{2}(26{)}^{2}+{c}^{26}{C}_{3}(-2{)}^{3}] \ =-52a+1300b-20800c \ \Rightarrow b-52c=-56 \ \Rightarrow b-52c=-56 \ a-52b+1300c=0\, \Rightarrow a=1300 \ \, -a+25b-400c=0\, b=100 \ c=3 \ \Rightarrow a+b+c=1403 \end{gathered}$

Solutions · 1