Question

Find the $4^{th}$ term in the expansion of $(x-2y{)}^{12}$

Answer 1

It is known that $(r+1)$th term $\left(T_{r+1}\right)$ in the binomial expansion of
$(a+b{)}^n$ is given by $T_{r+1}{=}^n{C}_r{b}^{n-r}{b}^r$
Thus the 4th term in the expansion of $(x+2y{)}^{12}$ is
$T_4=T_{3+1}{=}^{12}{C}_3\left(x{)}^{12-3}\right(-2y{)}^3=(-1{)}^3\frac{12!}{3!9!}.x^9.(2{)}^3.y^3$
$=-\frac{\mathrm{12.11.10}}{3.2}.(2{)}^3{x}^9{y}^3=-1760x^9{y}^3$