If the coefficients of x and x 2 in the expansion of 1 + x p 1 - x q , p , q ≤ 15 , are - 3 and - 5…

Given coefficient of $x$ in expansion of ${\left(1+x\right)}^p{\left(1-x\right)}^q$ is $-3$

$\Rightarrow {}^{q}C_0{}^{p}C_1-{}^{p}C_0{}^{q}C_1=-3$

$\Rightarrow p-q=-3 \cdots \left(1\right)$

Now coefficients of $x^2$ in expansion of ${\left(1+x\right)}^p{\left(1-x\right)}^q$

$-{}^{P}C_1^q{C}_1+{}^{q}C_0{}^{P}C_2+{}^{P}C_0{}^{q}C_2=-5$

$\Rightarrow -pq+\frac{p\left(p-1\right)}{2}+\frac{q\left(q-1\right)}{2}=-5 \cdots \left(2\right)$

Solving $\left(1\right)$ and $\left(2\right)$

$p=8,q=11$

Coefficient of $x^3$ in expansion of ${\left(1+x\right)}^p{\left(1-x\right)}^q$ will be,

$-{}^{q}C_3+{}^{p}C_3+{{}^{p}C}_1^q{C}_2-{{}^{p}C}_2^q{C}_1$

$=-{}^{11}C_3+{}^{8}C_3+{}^{8}C_1{}^{11}C_2-{}^{8}C_2{}^{11}C_1$

$=23$