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Sum of Coefficients of All Terms

If ∑r=0nCrx...

Question

If $n \sum r = 0 C_{r} x^{r}$ , then the value of $a C_{0} - (a + d) C_{1} + (a - 2 d) C_{2} - (a - 3 d) C_{3} + \dots + {(- 1)}^{n} (a - n d) C_{n}$ , equals

A

0

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B

(2 a - n) 2^{n} {(- 1)}^{n}

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C

{(- 1)}^{n} (a + n d) 2^{n}

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D

None of these

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Solution

The correct option is A $0$
Simplifying, we get
$= a (^{n} C_{0} -^{n} C_{1} +^{n} C_{2} + . . . (- 1)^{n}^{n} C_{n}) - d (^{n} C_{1} - 2^{n} C_{2} + 3^{n} C_{3} + . . . (- 1)^{n} d n^{n} C_{n})$
$= [a (1 - x)^{n} + d \frac{d (1 - x)^{n}}{d x}] |_{x = 1}$
$= 0$
Hence answer is $0.$

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