Mathematics - Binomial Theorem Question with Solution | TestHub

$(1+x{)}^m(1-x{)}^n=\left(1+m{C}_{1}x+m{C}_2{x}^2+\dots \dots \dots +m{C}_m{x}^m\right)$

$$\left(1-n{C}_{1}x+n{C}_2{x}^2-\dots .\right)$$

Coefficient of $x=m{C}_1-n{C}_1=3$.

Therefore, $m-n=3$.

Coefficient of $x^2=m{C}_2+n{C}_2-m{C}_1\cdot n{C}_1=-6$.

Therefore, $\frac{m(m-1)}{2}+\frac{n(n-1)}{2}-mn=-6$. From (1) and (2):

$\frac{(n+3)(n+2)}{2}+\frac{n(n-1)}{2}-n(n+3)=-6$

$$n^2+5n+6+n^2-2n^2-6n=-12$$

$$-2n=-18$$

$$n=9$$

Therefore, $m=n+3=12$.