Question

If $f\left(x\right)=1-x+x^2-x^3+....-x^{15}+x^{16}+x^{17}$, then the coefficient of $x^2$ in $f(x-1)$ is?

• A. $826$
• B. $816$
• C. $822$
• D. None of these

Answer 1

Answer: (B)

$816$

$f\left(x\right)=1-\left(x-1\right)+{\left(x-1\right)}^2-{\left(x-1\right)}^3+{\left(x-1\right)}^4....-{\left(x-1\right)}^{15}+{\left(x-1\right)}^{16}+{\left(x-1\right)}^{17}$

The coefficient of $x^2$ by using binomial expansion

${}^2{c}_0{x}^2{+}^3{c}_1{x}^2{+}^4{c}_2{x}^2{+}^5{c}_3{x}^2{+}^6{c}_4{x}^2{+}^7{c}_5{x}^2{+}^8{c}_6{x}^2{+}^9{c}_7{x}^2$

${+}^{10}{c}_8{x}^2{-}^{11}{c}_9{x}^2{+}^{12}{c}_{10}{x}^2{-}^{13}{c}_{11}{x}^2{+}^{14}{c}_{12}{x}^2{-}^{15}{c}_{13}{x}^2{+}^{16}{c}_{14}{x}^2{-}^{17}{c}_{15}{x}^2$

$\Rightarrow x^2\left(1+3+6+10+15+21+28+36+45+55+66+78+91+105+120+136\right)$

= 372 + 444 = 816