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Q.
Find the coefficient of $x^6y^3$ in the expansion of $(x + 2y)^9$.
Binomial Theorem
Solution:
Let $x^6y^3$ be the $(r + 1)^{th}$ term in the expansion of $(x + 2y)^9$.
Now $T_{r+1} = \,{}^{9}C_{r}x^{9-r}\left(2y\right)^{r}$
$= \,{}^{9}C_{r}2^{r}\cdot x^{9-r}\cdot y^{r}$.
Comparing the indices of $x$ and $y$ in $x^{6}y^{3}$ and in $T_{r+1}$, we get $r = 3$.
Thus, the coefficient of $x^{6}y^{3} = \,{}^{9}C_{3}2^{3} = 672$.