Find the coefficient of $x^{9}$ in $(1 + 3 x + 3 x^{2} + x^{3})^{15}$ .

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Solution

$(1 + 3 x + 3 x^{2} + x^{3})^{15} = ((1 + x)^{3})^{15}$

$= (1 + x)^{45}$

$= 1 + 45_{C_{1}} x + 45_{C_{2}} x^{2} + . . . . . .$

So,the coefficient of $x^{k}$ in $(1 + x)^{n}$ in $n_{C_{K}}$

$∴$ the coefficient of $x^{9}$ in $(1 + 3 x + 3 x^{2} + x^{3})^{15}$ is

$45_{C_{9}}$

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