If the coefficients of x and x ^ 2 in ( 1 + x ) ^ p ( 1 - x ) ^ q...

Given that The coefficient of $x$ and ${x}^{2} in${\left(1 + x\right)}^{p}{\left(1 - x\right)}^{q} are $4$ and $- 5$.

$⇒ {\left(1 + x\right)}^{p}{\left(1 - x\right)}^{q} = \left(1 + px + \dfrac{p\left(p - 1\right)}{2 !}{x}^{2} + . . . .\right)\left(1 - qx + \dfrac{q\left(q - 1\right)}{2 !}{x}^{2} - . . . .\right)$

Now coefficient of $x$ from the above expansion will be

$p - q$ which is equal to $4$

$⇒ p - q = 4$.

Similarly coefficient of ${x}^{2} is$- 5$.</p><p>$⇒ \dfrac{p\left(p - 1\right)}{2} + \dfrac{q\left(q - 1\right)}{2} - pq = - 5$</p><p>$⇒ \dfrac{{p}^{2} - 2pq + {q}^{2}}{2} - \dfrac{\left(p + q\right)}{2} = - 5$</p><p>$⇒ \dfrac{{\left(p - q\right)}^{2}}{2} - \dfrac{\left(p + q\right)}{2} = - 5$</p><p>$⇒ \dfrac{16}{2} + 5 = \dfrac{\left(p + q\right)}{2}$</p><p>$⇒ p + q = 26$and$p - q = 4$</p><p>On solving the above equations we get,</p><p>$p = 15$and$q = 11$.</p><p>$⇒ 2p + 3q = 2\left(15\right) + 3\left(11\right) = 63$</p><p>Therefore, the required answer is$63.