Question

If in the expansion of $(1+x{)}^m\cdot (1-x{)}^n$, the coefficients of $x$ and $x^2$ are $3$ and $-6$ respectively, then

• A. $m=6,n=15$
• B. $m=9,n=12$
• C. $m=12,n=9$
• D. $m=24,n=21$

Answer 1

The correct option is C $m=12,n=9$

$(1+a_{1}x+a_2{x}^2+...)(1-p_{1}x+p_2{x}^2+...)$
Hence coefficient of x will be
$=-p_1+a_1$
$=-{}^n{C}_1+{}^m{C}_1$
$=m-n$
$=3$ ...(i)
Coefficient of $x^2$ will be
$=p_2-a_1{p}_1+a_2$
$={}^n{C}_2-{}^n{C}_1{}^n{C}_2+{}^m{C}_2$
$=\frac{n(n-1)}{2}-mn+\frac{m(m-1)}{2}$
$=-6$
Hence
$n^2-n-2mn+m^2-m=-12$
$n^2-2mn+m^2-(m+n)=-12$
$(m-n{)}^2-(m+n)=-12$
$9-(m+n)=-12$ ...($m-n=3$)
$m+n=21$ ...(ii)
Adding i and ii, we get
$m=12$ hence $n=9$.