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

(a x^2+b c+c) sum _ r=0^26 ^26 C_ r(-2 x)^ r Coeff. of x^2: a . ^26 C_0(-2)^0+ b . ^26 C_1(-2)+ c. ^26 C_2(-2)^2=0 => a-52 b+1300 c=0 ......(1) Coeff. of x^3: a . ^26 C_1(-2)+ b . ^26 C_2(-2)^2+ c. ^26 C_3(-2)^3=0 => -52 a+1300 b-20800 c=0 .....(2) Coeff. of x=-56 => b . ^26 C_0(-2)^0+ c . ^26 C_1(-2)^1=-56 b-52 c=-56 .....(3) After solving (1), (2) (3) \ a=1300, b=100, c=3 => a+ b+ c=1403

If the coefficient of x in the expansion of a x 2 b x c... | null (Mathematics)

\left(a x^{2}+b c+c\right) \sum_{\mathrm{r}=0}^{26}{ }^{26} \mathrm{C}_{\mathrm{r}}(-2 \mathrm{x})^{\mathrm{r}}\\

Coeff. of

\mathrm{x}^{2}: \mathrm{a} .{ }^{26} \mathrm{C}_{0}(-2)^{0}+\mathrm{b} .{ }^{26} \mathrm{C}_{1}(-2)+

{ }^{26} \mathrm{C}_{2}(-2)^{2}=0\\

\Rightarrow a-52 b+1300 c=0 \tag......{(1)}\\

Coeff. of

\mathrm{x}^{3}: \mathrm{a} .{ }^{26} \mathrm{C}_{1}(-2)+\mathrm{b} .{ }^{26} \mathrm{C}_{2}(-2)^{2}+

{ }^{26} \mathrm{C}_{3}(-2)^{3}=0\\

\Rightarrow-52 a+1300 b-20800 c=0 \tag.....{(2)}\\

Coeff. of

x=-56\\

\Rightarrow \mathrm{b} .{ }^{26} \mathrm{C}_{0}(-2)^{0}+\mathrm{c} .{ }^{26} \mathrm{C}_{1}(-2)^{1}=-56\\

b-52 c=-56 \tag.....{(3)}\\

After solving (1), (2) & (3)

\\ \mathrm{a}=1300, \mathrm{~b}=100, \mathrm{c}=3\\

\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=1403

Explanation

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