Write the coefficient of

To find: the coefficient of $x^{7} y^{2}$ in the expansion of $(x+2 y)^{9}$

Formula Used:

A general term, $T_{r+1}$ of binomial expansion $(x+y)^{n}$ is given by,

$T_{r+1}={ }^{n} C_{r} x^{n-r} y^{r}$ where

${ }^{n} C_{r}=\frac{n !}{r !(n-r) !}$

Now, finding the general term of the expression, $(x+2 y)^{9}$, we get

$T_{r+1}={ }^{9} C_{r} \times x^{9-r} \times(2 y)^{r}$

The value of $r$ for which coefficient of $x^{7} y^{2}$ is defined

$R=2$

Hence, the coefficient of $x^{7} y^{2}$ in the expansion of $(x+2 y)^{9}$ is given by:

$T_{3}={ }^{9} C_{3} \times x^{9-2} \times(2 y)^{2}$

$T_{3}={ }^{9} C_{3} \times 4 \times x^{7} \times(y)^{2}$

$T_{3}=\frac{9 !}{3 ! \times 6 !} \times 4 \times x^{7} \times(y)^{2}$

$T_{3}=\frac{9 \times 8 \times 7 \times 6 !}{6 \times 6 !} \times 4 \times x^{7} \times(y)^{2}$

$T_{3}=336$

Thus, the coefficient of $x^{7} y^{2}$ in the expansion of $(x+2 y)^{9}$ is 336 .