If the coefficients of x and x^2 in the expansion of (1+x)^p(1-x)^q, p,q15, are -3 and -5 …

Question

If the coefficients of $x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q$, $p,q\le15$, are $-3$ and $-5$ respectively, then the coefficient of $x^3$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 23

Accepted Answer

Coefficient of $x$: $p-q=-3$

Coefficient of $x^2$: $(p-q)^2-(p+q)=-10\Rightarrow9-(p+q)=-10\Rightarrow p+q=19$

With $p-q=-3$: $q=11,\ p=8$.

Coefficient of $x^3$ in $(1+x)^8(1-x)^{11}$: $-{}^{11}C_3+{}^8C_1{}^{11}C_2-{}^8C_2{}^{11}C_1+{}^8C_3=23$

Solutions · 1