The coefficient of $x^{7}$ in the polynomial expansion of ${(1 + 2 x - x^{2})}^{4}$ is

- 8

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12

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6

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Solution

The correct option is C $- 8$
Given
$(1 + 2 x - x^{2})^{4}$
General term given expansion
$T_{r + 1} = \sum_{r = 0}^{4}^{4} C_{r} (x (2 - x))^{r}$
When we put $r = 4$ we get term having $x^{7}$
$T_{3 + 1} =^{4} C_{4} x^{4} (2 - x)^{4}$
We can written as
$T_{4} = x^{4} \sum_{r = 0}^{4}^{4} C_{r} (2)^{4 - r} x^{r} (- 1)^{r}$
Here on putting $r = 3$
$T_{4} =^{4} C_{3} (2)^{1} x^{3} (- 1)^{3} x^{4}$

$T_{4} = - 2^{4} C_{3} x^{7}$
Coefficient of above term is
$- 2 C_{3}^{4} = - 2 \frac{4!}{1! 3!} = - 8$

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