Question

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

Answer 1

Answer: 23

Since coefficient of $x$ is $-3$
$\Rightarrow {}^p{C}_1-{}^q{C}_1=-3$
$\Rightarrow p-q=-\mathrm{3.....}$(1)
Comparing coefficients of $x^2$
$-{}^p{C}_1{}^q{C}_1+{}^p{C}_2+{}^q{C}_2=-\mathrm{5.......}$(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
$-{}^q{C}_3+{}^p{C}_3+{}^p{C}_1{}^q{C}_2-{}^p{C}_2{}^q{C}_1$
$=-{}^{11}{C}_3+{}^8{C}_3+{}^8{C}_1{}^{11}{C}_2-{}^8{C}_2{}^{11}{C}_1$
$=23$