For natural numbers m, n if `(1-y)^(m)(1+y)^(n) = 1+a_(1)y+a_(2)y^(2) + "……."` and `a_(1) = a_(2) = 10`, then `(m,n)` is :

Options

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

Answer

d

Explanation

We have,
`(1 - y)^(m) (1 + y)^(n)`
`(""^(m)C_(0) - ""^(m)C_(1)y ^(2) -...)xx(""^(n)C_(0) + ""^(n)C_(1) y + ""^(n)C_(2) y^(2) +...)`
=` 1 + (n+ m) y + {(m(m-1) + n (n-1))/(2) -mn }y^(2) +…`
`therefore a_(1) = n - m and a_(2) = (m (m -1) + n(n-1))/(2) - mn`
`rArr n-m = 10 and (m^(2) + n^(2) - (m+n ) - 2mn)/(2) = 100 [ because a_(1) = a_(2) = 10]`
`rArr n-m = -10 and (m - n)^(2) - (m +n) = 20`
`rArr m - n = - 10 and m + n = 80`
`rArr m = 35 and n = 45` .