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

930

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715

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672

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750

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Solution

The correct option is C $672$
General term of expansion $(a + b)^{n}$ is
$T_{r + 1} = {n_{C}}_{r} a^{n - r} b^{r}$
For $(x + 2 y)^{9}$ ,
Putting n =9, a=x, b=2y
$T_{r + 1} = {9_{C}}_{r} (x)^{9 - r} (2 y)^{r}$
$T_{r + 1} = {9_{C}}_{r} (x)^{9 - r} . (y)^{r} . (2)^{r}$
Comparing with $x^{6} y^{3}$ , we get,
$r = 3$
Therefore,
$T_{r + 1} = {9_{C}}_{3} (x)^{9 - 3} . y^{3} {.2}^{3}$

$= \frac{9!}{3! .6!} (2)^{3} \times x^{6} \times y^{3}$

$= 672 x^{6} y^{3}$
Hence, coefficient of $x^{6} y^{3}$ is $672$ .

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