If in the expansion of ` (1 +x)^(m) (1 - x)^(n)` , the coefficients
of x and ` x^(2) ` are 3 and - 6 respectively, the value of m and n are

Answer

c d

Explanation

We have,
`(1 + x)^(m) (1 -x)^(n)`
`(""^(m)C_(0) + ""^(m)C_(1) x + ""^(m)C_(2) x^(2) +...+ ""^(m)C_(m) x^(m))`
xx`(""^(n)C_(0) + ""^(n)C_(1) x + ""^(n)C_(2) x^(2) +...+(-1)^(n) ""^(n)C_(n) x^(n))`
`""^(m)C_(0) ""^(n)C_(0) -( ""^(m)C_(0) ""^(n)C_(1)- ""^(n)C_(0) ""^(m)C_(1))x`
`+(""^(m)C_(0) ""^(n)C_(2) + ""^(n)C_(0) ""^(m)C_(2)- ""^(m)C_(1) ""^(n)C_(1))x^(2)` +.....
It is given that the coefficients of x and `x^(2)` in the expansion of
`(1+x)^(m) (1-x)^(n)` are 3 and -6 respectively.
`therefore - ""^(m)C_(c) ""^(n)C_(1) - ""^(n)C_(0) ""^(m)C_(1)) = 3`
and , `""^(m)C_(0) ""^(n)C_(2) + ""^(n)C_(0) ""^(m)C_(2) - ""^(m)C_(1) ""^(n)C_(1) = -6`
`rArr m - n = 3 and n(n -1) +m (m-1) -2mn = -12`
`rArr m - n = 3 and (m-n)^(2) - (m +n) = -12`
` rArr m - n = 3 and m+n = 21 `
`rArr m = 12 , n=9` .