Question

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

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

Answer 1

Answer: (D)

Given function is
$f(y)=1-(y-1)+(y-1{)}^2-(y-1{)}^3+........-(y-1{)}^{17}$
In the expansion of $(y-1{)}^n$
$T_{r+1}={}^n{C}_r{y}^{n-r}{\left(-1\right)}^r$
coeff of $y^2$ in $(y-1{)}^2={}^2{C}_0$
coeff of $y^2$ in $(y-1{)}^3={}^{-3}{C}_1$
coeff of $y^2$ in $(y-1{)}^4={}^4{C}_2$
So, coeff of termwise is
${}^2{C}_0{+}^3{C}_1{+}^4{C}_2{+}^5{C}_3+.......{+}^{17}{C}_{15}$
$=1{+}^3{C}_1{+}^4{C}_2{+}^5{C}_3+.......{+}^{17}{C}_{15}$
$=\left({}^3{C}_0{+}^3{C}_1\right){+}^4{C}_2{+}^5{C}_3+.......{+}^{17}{C}_{15}$
${=}^4{C}_1{+}^4{C}_2{+}^5{C}_3+.......{+}^{17}{C}_{15}$
${=}^5{C}_2{+}^5{C}_3+.......{+}^{17}{C}_{15}$
${=}^{18}{C}_{15}{=}^{18}{C}_3$