Question

For natural numbers m, n if $(1- y)^m(1+ y)^n$
$= 2 1+ a_1y + a_2 y^2 + ...$ and $a_1 = a_2 = 10$, then (m, n) is

• A. (20, 45)
• B. (35, 20)
• C. (45, 35)
• D. (35, 45)

Answer 1

Answer: (D)

$\left(1- y\right)^{m}\left(1+ y\right)^{n}$
$= \left[1-^{m}C_{1}y +^{m}C_{2} y^{2}- ..... \right] \left[1 + ^{n}C_{1}y + ^{n}C_{2} y^{2}+ .....\right]$
$= 1 + \left(n-m\right)y +\left\{\frac{m\left(m-1\right)}{2}+\frac{n\left(n-1\right)}{2}-mn\right\}y^{2} + .....$
By comparing coefficients with the given expression, we get
$\therefore a_{1} = n - m =10$ and
$a_{2} =\frac{ m^{2}+n^{2}-m-n-2mn}{2} = 10$
So, $n - m = 10$ and $\left(m - n\right)^{2} - \left(m+ n\right) = 20$
$\Rightarrow m+ n = 80\quad\therefore m = 35, n = 45$