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

Options

A. 3
B. 6
C. 9
D. 12

Answer

c,d

Explanation

We have,
Coefficient of x in ` (1 + x)^(m) (1-x)^(n) = ""^(m)C_(1) - ""^(n)C_(1)`
and coefficient of ` x^(2) "in" (1 + x)^(m) (1 - x)^(n) = ""^(m)C_(2) - ""^(m)C_(1) ""^(n)C_(1) + ""^(n)C_(2)`
According to the question , ` ""^(m)C_(1) - ""^(n)C_(1) = 3`
` rArr m - n = 3` ...(i)
and ` ""^(m)C_(2) - ""^(m)C_(1) ""^(n)C_(1) + ""^(n)C_(2) = - 6`
` rArr (m(m-1))/(2) - mn + (n(n-1))/(2) = - 6`
` rArr (m-n)^(2) - (m +n) = -12`
` rArr 9 - (m+ n) = 12 " "` [ from Eq. (i)]
or m + n= 21 ....(ii)
From Eqs . (i) and (ii) , we get
m = 12 and n = 9