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 $(y - 1)^2 = {^2C_0}$
coeff of $y^2$ in $(y - 1)^3 = {^{-3}C_1}$
coeff of $y^2$ in $(y - 1)^4 = {^4C_2}$
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} $