A) 6
B) 9
C) 12
D) 24
Correct Answer: C
Solution :
Given, \[{{(1+x)}^{m}}{{(1-x)}^{n}}\] \[=\left[ 1+mx+m\frac{(m-1)}{2!}{{x}^{2}}+.... \right]\] \[\left[ 1-nx+\frac{n(n-1)}{2!}{{x}^{2}}-.... \right]\] \[=1+(m+n)x+\left[ \frac{{{n}^{2}}-n}{2}-mn+\frac{{{m}^{2}}-m}{2} \right]{{x}^{2}}\] \[+..............\] Also, given \[m-n=3\Rightarrow m=m-3\] and\[\frac{{{n}^{2}}-n}{2}-mn+\frac{{{m}^{2}}-m}{2}=-6\] \[\Rightarrow \] \[\frac{(m-3)(m-4)}{2}-m(m-3)\] \[+\frac{{{m}^{2}}-m}{2}=-6\] \[\Rightarrow \]\[{{m}^{2}}-7m+12-2{{m}^{2}}+6m+{{m}^{2}}-m+12=0\] \[\Rightarrow \] \[-2m+24=0\] \[\Rightarrow \] \[m=12\]
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