For natural numbers `m`, `n`, if `(1-y)^m(1+y)^n=1+a_1y+a_2y^2+...,` and `a_1=a_2=10` then (A)`mltn` (B)`mgtn` (C)`m+n=80` (D)`m-n=20`

Options

A. `m lt n`
B. `m gt n`
C. `m+n=80`
D. `m-n=20`

Answer

A, C

Explanation

`(1-y)^(m)(1+y)^(n)`
`= (1-.^(m)C_(1)y+.^(m)C_(2)y^(2) - "……") (1+.^(n)C_(1)y+.^(n)C_(2)y^(2) + "…..")`
`= 1+(n-m)y+{(m(m-1))/(2) + (n(n-1))/(2)-mn} y^(2) + "….."`
Given,
`a_(1) = 10`
`rArr a_(1) = n - m = 10" " (1)`
`a_(2) = (m^(2) + n^(2) - m - n - 2mn)/(2) = 10`
`(m-n) = 80" "(2)`
Solving (1) and (2), we get `m = 35, n = 45`