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

6 n_{C_{3 n}} (- x)^{3 n}

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6 n_{C_{2 n}} (- x)^{2 n + 1}

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4 n_{C_{3 n}} (- x)^{3 n}

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6 n_{C_{3 n}} (- x)^{3 n - 1}

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Solution

The correct option is D $6 n_{C_{3 n}} (- x)^{3 n}$
Given, $(1 - 3 x + 3 x^{2} - x^{3})^{2 n}$

$= [(1 - x)^{3}]^{2 n}$

$= (1 - x)^{6 n}$

Middle term $= \frac{N}{2} + 1$

$= \frac{6 n}{2} + 1$

$= (3 n + 1)^{t h}$ term

Now consider the following,

$T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$ ...(i)

Where $T$ represents the term

Here the middle term is the $(3 n + 1)^{t h}$ term.

Hence, $r + 1 = 3 n + 1$

$r = 3 n$

Substituting in (i), we get

$(- 1)^{3 n}^{6 n} C_{3 n} 1^{6 n - 3 n} x^{3 n}$

$= (- 1)^{3 n}^{6 n} C_{3 n} x^{3 n}$

$=^{6 n} C_{3 n} (- x)^{3 n}$

Hence answer is A

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