Find the coefficient of $t^{8}$ in the expansion of $(1 + 2 t^{2} - t^{3})^{9}$ .

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Solution

$t^{8}$ can be obtained in the following ways using multinomial theorem.
$(- t^{3})^{2} (2 t^{2})^{1} . (1)^{6}$ Coefficient is $= 2 \times \frac{9!}{2! .1! .6!} = 504$ .
$(- t^{3})^{0} (2 t^{2})^{4} . (1)^{5}$ Coefficient is $= 2^{4} \times \frac{9!}{4! .0! .5!} = 2016$ .
Hence the coefficient of $t^{8}$ is $\to$ $504 + 2016 = 2520$

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