The middle term in the expansion of $(1 - 3 x + 3 x^{2} - x^{3})^{6}$ is

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Solution

$\begin{matrix} w e h a v e, (1 - 3 x + 3 x^{2} - x^{3})^{6} = ((1 - x)^{3}) = (1 - x)^{18} (1 - x)^{n} \to n + 1 m i d d l e : {(\frac{n}{2} + 1)}^{t h} \Rightarrow^{n} c_{0} -^{n} c_{1} x = (- 1)^{n}^{n} c_{n} \times n \Rightarrow (- 1)^{r}^{n} c_{r} x^{r} [r = 9] \Rightarrow (- 1)^{9}^{18} c_{9} x^{9} ∴^{18} c_{9} (- x)^{9} \end{matrix}$

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Middle term in the expansion of $(1 + 3 x + 3 x^{2} + x^{3})^{6}$ is

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