Find the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$
Find the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$
It is known $(r+1)^{\text {th }}$ term, $T_{r+1},$ in the binomial expansion of $(a+b)^{n}$ is given by ${T_{r + 1}} = {\,^n}{C_r}{a^{n - r}}{b^r}$
Thus, the $4^{\text {th }}$ term in the expansion of $\left(x^{2}-2 y\right)^{12}$ is
${T_4} = {T_{3 + 1}} = {\,^{12}}{C_3}{(x)^{12 - 3}}{( - 2y)^3} = {( - 1)^3} \cdot \frac{{12!}}{{3!9!}} \cdot {x^9} \cdot {(2)^3} \cdot {y^3}$
$=-\frac{12 \cdot 11 \cdot 10}{3 \cdot 2} \cdot(2)^{3} x^{9} y^{3}=-1760 x^{9} y^{3}$
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