If $n > 1$ then value of the expression is
$C_{0} - 2 C_{1} + 3 C_{2} - 4 C_{3} . . . . . . + {(- 1)}^{n} (n + 1) C_{n}$ is

- 1

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none of these

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Solution

The correct option is B $0$
We have
$C_{0} x + C_{1} x^{2} + C_{2} x^{3} + . . . . . . + C_{n} x^{n + 1} = x {(1 + x)}^{n}$
Differentiating both the sides, we get
$C_{0} + 2 C_{1} x + 3 C_{2} x^{2} + . . . . . + (n + 1) C_{n} x^{n} = {(1 + x)}^{n} + n x {(1 + x)}^{n - 1} . . . . . . (1)$
Substituting $x = - 1$ , we get
$C_{0} - 2 C_{1} + 3 C_{2} - 4 C_{3} . . . . . . + {(- 1)}^{n} (n + 1) C_{n} = 0$

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C_{0} - 2 C_{1} + 3 C_{2} - 4 C_{3} + . . . . + (- 1)^{n} (n + 1) C_{n} = 0

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