Find the coefficient of $x^{6} y^{3}$ in the expansion of $(x+2 y)^{9}$
Find the coefficient of $x^{6} y^{3}$ in the expansion of $(x+2 y)^{9}$
Suppose $x^{6} y^{3}$ occurs in the $(r+1)^{\text {th }}$ term of the expansion $(x+2 y)^{9}$
Now ${T_{r + 1}} = {\,^9}{C_r}{x^{9 - r}}{(2y)^r} = {\,^9}{C_r}{2^r} \cdot {x^{9 - r}} \cdot {y^r}$
Comparing the indices of $x$ as well as $y$ in $x^{6} y^{3}$ and in $T_{r+1},$ we get $r=3$
Thus, the coefficient of $x^{6} y^{3}$ is
${\,^9}{C_3}{2^3} = \frac{{9!}}{{3!6!}} \cdot {2^3} = \frac{{9.8.7}}{{3.2}} \cdot {2^3} = 672$
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