Question:
If in the expansion of $(1 + x)^m (1 - x)^n$, the coefficients of $x$ and $x^2$ are 3 and - 6 respectively, then m is euqal to
If in the expansion of $(1 + x)^m (1 - x)^n$, the coefficients of $x$ and $x^2$ are 3 and - 6 respectively, then m is euqal to
Updated On: Jun 14, 2022
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The Correct Option is C
Solution and Explanation
$(1+x)^m(1-x)^n= \bigg[1+mx+\frac{m(m-1)}{2}x^2+...\bigg]$
$\hspace30mm \, \bigg[1-nx+\frac{n(n-1)}{2}x^2+...\bigg]$
=1+(m-n)x+$\bigg[ \frac{m(m-1)}{2}+\frac{n(n-1)}{2}-mn\bigg] x^2+...$
term containing power of x$\ge $ 3.
Now, m -n = 3 $\hspace20mm $....(i)
and $ \, \, \, \frac{1}{2}m(m-1)x+\frac{1}{2}n(n-1)-mn=-6$
$\Rightarrow \, \, \, \, \, \, m(m-1)+n(n-1)-2mn=-12 $
$\Rightarrow \, \, \, \, \, \, m^2-m+n^2-n-2mn=-12$
$\Rightarrow \, \, \, \, \, \, (m-n)^2-(m+n)=-12 $
$\Rightarrow \, \, \, \, \, \, \, \, \, \, \, \, m+n=9+12=21 \, \, \, \, \, \, .....(ii)$
On solving Eqs. (i) and (ii), we get m = 12
$\hspace30mm \, \bigg[1-nx+\frac{n(n-1)}{2}x^2+...\bigg]$
=1+(m-n)x+$\bigg[ \frac{m(m-1)}{2}+\frac{n(n-1)}{2}-mn\bigg] x^2+...$
term containing power of x$\ge $ 3.
Now, m -n = 3 $\hspace20mm $....(i)
and $ \, \, \, \frac{1}{2}m(m-1)x+\frac{1}{2}n(n-1)-mn=-6$
$\Rightarrow \, \, \, \, \, \, m(m-1)+n(n-1)-2mn=-12 $
$\Rightarrow \, \, \, \, \, \, m^2-m+n^2-n-2mn=-12$
$\Rightarrow \, \, \, \, \, \, (m-n)^2-(m+n)=-12 $
$\Rightarrow \, \, \, \, \, \, \, \, \, \, \, \, m+n=9+12=21 \, \, \, \, \, \, .....(ii)$
On solving Eqs. (i) and (ii), we get m = 12
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.