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Mathematics Question on Binomial theorem

If in the expansion of (1+x)m(1−x)n(1 + x)^m (1 - x)^n, the coefficients of xx and x2x^2 are 3 and - 6 respectively, then m is euqal to

A

6

B

9

C

12

D

24

Answer

12

Explanation

Solution

(1+x)m(1−x)n=[1+mx+m(m−1)2x2+...](1+x)^m(1-x)^n= \bigg[1+mx+\frac{m(m-1)}{2}x^2+...\bigg]
\hspace30mm \, \bigg[1-nx+\frac{n(n-1)}{2}x^2+...\bigg]
=1+(m-n)x+[m(m−1)2+n(n−1)2−mn]x2+...\bigg[ \frac{m(m-1)}{2}+\frac{n(n-1)}{2}-mn\bigg] x^2+...
term containing power of x≥\ge 3.
Now, m -n = 3 \hspace20mm ....(i)
and    12m(m−1)x+12n(n−1)−mn=−6 \, \, \, \frac{1}{2}m(m-1)x+\frac{1}{2}n(n-1)-mn=-6
⇒      m(m−1)+n(n−1)−2mn=−12\Rightarrow \, \, \, \, \, \, m(m-1)+n(n-1)-2mn=-12
⇒      m2−m+n2−n−2mn=−12\Rightarrow \, \, \, \, \, \, m^2-m+n^2-n-2mn=-12
⇒      (m−n)2−(m+n)=−12\Rightarrow \, \, \, \, \, \, (m-n)^2-(m+n)=-12
⇒            m+n=9+12=21      .....(ii)\Rightarrow \, \, \, \, \, \, \, \, \, \, \, \, m+n=9+12=21 \, \, \, \, \, \, .....(ii)
On solving Eqs. (i) and (ii), we get m = 12