The coefficient of $x^{13}$ in ${{(1 - x)}^{5} (1 + x + x^{2} + x^{3})}^{4}$ is

4

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Solution

The correct option is A $4$
$\to (1 - x)^{5} (1 + x + x^{2} (1 + x))^{4}$
$= (1 - x)^{5} (1 + x)^{4} (1 + x^{2})^{4}$
$= (1 - x) (1 - x^{2})^{4} (1 + x^{2})^{4}$
$= (1 - x) (1 - x^{4})^{4} = (1 - x^{4})^{4} - x (1 - x^{4})^{4}$
For $x^{13}, (1 - x^{4})^{4}$ would have no terms (even power)
In second term, general term is
$- x \times^{4} C_{r} (- x^{4})^{r}$
$\Rightarrow r = 3$
$\Rightarrow$ coefficient $=^{- 4} C_{3} (- 1)^{3} =^{4} C_{3} = 4$
$\Rightarrow (A)$

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