Question

If the coefficients of $x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q, p, q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x^3$ is equal to ________.

Answer 1

Since coefficient of $x$ is $-3$
$ \Rightarrow{ }^{ p } C _1-{ }^{ q } C _1=-3$
$\Rightarrow p - q =-3 .....$(1)
Comparing coefficients of $x ^2$
$ -{ }^{ p } C _1{ }^{ q } C _1+{ }^{ p } C _2+{ }^{ q } C _2=-5 .......$(2)
$ - pq +\frac{ p ( p -1)}{2}+\frac{ q ( q -1)}{2}=-5$
Solving (1) and (2)
$p =8, q =11$
Coefficient of $x ^3$ is
$ -{ }^{ 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 $
$=23$