Explanation

(c)
${(1 + x)}^{m}{(1 - x)}^{n} = \left( 1 + mx + \frac{m(m - 1)x^{2}}{2!} + ... \right)$
$\left( 1 - nx + \frac{n(n - 1)}{2!}x^{2} - ... \right)$
$= 1 + (m - n)x + \left\lbrack \frac{n^{2} - n}{2} - mn + \frac{\left( m^{2} - m \right)}{2} \right\rbrack x^{2}...$
Given, $m - n = 3\ \Longrightarrow n = m - 3$
and $\frac{n^{2} - n}{2} - mn + \frac{m^{2} - m}{2} = - 6$
$\Longrightarrow \ \frac{(m - 3)(m - 4)}{2} - m(m - 3) + \frac{m^{2} - m}{2} = - 6$
$\Longrightarrow \ m^{2} - 7m + 12 - {2m}^{2} + 6m + m^{2} - m + 12 = 0$
$\Longrightarrow \ - 2m + 24 = 0\ \Longrightarrow \ \ m = 12$