Question

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

• A. ${}^{17}{C}_3$
• B. ${}^{18}{C}_3$
• C. ${}^{17}{C}_2$
• D. ${}^{18}{C}_2$

Answer 1

The correct option is B ${}^{18}{C}_3$

$f\left(y\right)1=1-\left(y-1\right)+{\left(y-1\right)}^2-{\left(y-1\right)}^3+..........{\left(y-1\right)}^{17}Then,Coefficientof{y}^2{\Rightarrow }^2{C}_0{y}^2{+}^3{C}_1{y}^2{+}^4{C}_2{y}^2+.........{.}^{17}{C}_{15}{y}^2{\Rightarrow }^2{C}_0{+}^3{C}_1{+}^4{C}_2+.........{.}^{17}{C}_{15}\left[{\because }^n{C}_r{=}^n{C}_{n-r}\right]\Rightarrow {}^2{C}_0{+}^3{C}_2{+}^4{C}_2{+}^5{C}_2.........{.}^{17}{C}_2$