If the sum of the coefficients of $x^{2}$ and coefficients of $x$ in the expansion of $(1 + x)^{m} (1 - x)^{n}$ is equal to $- m$ , then the value of $3 (n - m)$ is
(Note : $m, n$ are distinct )

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Solution

$(1 + x)^{m} (1 - x)^{n}$
Expanding using Binomial Theorem we get,
$(1 + m x +^{m} C_{2} x^{2} + . . . +^{m} C_{m} x^{m}) (1 - n x +^{n} C_{2} x^{2} . . . + (- 1)^{n}^{n} C_{n} x^{n})$
Therefore coefficient of $x$ is $m - n$ ...(i)
coefficient of $x^{2}$ is $^{n} C_{2} - m n +^{m} C_{2}$ ..(ii)
Adding (I) and (ii), we get
$m - n + (\frac{n (n - 1)}{2} + \frac{m (m - 1)}{2} - m n) = - m$
$\Rightarrow (m - n) + \frac{1}{2} (n^{2} - 2 m n + m^{2} - (m + n)) = - m$
$\Rightarrow (m - n) + \frac{(m - n)^{2}}{2} = \frac{n - m}{2}$
$\Rightarrow 3 (m - n) + (m - n)^{2} = 0$
$\Rightarrow (m - n) (m - n + 3) = 0$
Since, $m$ and $n$ are distinct , $n - m = 3$
Hence, $3 (n - m) = 9$ .

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