Find the coefficient of $x^{6} y^{3}$ in the expansion of $(x + 2 y)^{9}$

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Solution

The general term in the expansion of $(x + 2 y)^{9}$ is given by
$T_{r + 1} =^{9} C_{r} \times x^{(9 - r)} \times (2 y)^{r} \Rightarrow T_{r + 1} =^{9} C_{r} \times 2^{r} \times x^{(9 - r)} \times y^{r}$
we have to find the coefficient of $x^{6} y^{3}$
$Putting r = 3 in (i), we get T_{(3 + 1)} =^{9} C_{3} \times 2^{3} \times (x^{6} y^{3}) ∴ coefficient of x^{6} y^{3} in the given expansion = (^{9} C_{3} \times 2^{3}) = (\frac{9 \times 8 \times 7}{3 \times 2 \times 1} \times 8) = 672.$
Hence the coefficient of $x^{6} y^{3}$ in the given expansion is 672.

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