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Combination

If C0 , C1,...

Question

If $C_{0}, C_{1}, C_{2}, . . . . . C_{n}$ are binomial coefficients of different terms in the expansion of $(1 + x)^{n}$ then $C_{0} + 2. C_{1} + 3. C_{2} + 4. C_{3} + . . . . + (1)^{n} . (n + 1) C_{n}$ equals

A

- n {.2}^{n - 1}

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B

0

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C

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

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D

none of these

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Solution

The correct option is B $0$
$T h e g e n e r a l t e r m i s g i v e n b y T r = {(- 1)}^{r} . (r + 1) C_{r} H e n c e t h e s e r i e s i s g i v e n b y \sum_{0}^{n} {(- 1)}^{r} . (r + 1) C_{r} = \sum_{0}^{n} {(- 1)}^{r} . r C_{r} + \sum_{0}^{n} {(- 1)}^{r} C_{r} S i n c e C_{0} - C_{1} + C_{2} - C_{3} + . . . . = 0 a n d r C_{r} = n C_{r - 1}^{n - 1} H e n c e u s i n g t h e a b o v e t w o r e l a t i o n w e c a n i m p l y t h a t t h e s u m o f a l l t h e t e r m s i n t h e s e r i e s w i l l b e c o m e 0$

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Similar questions

C_{0}, C_{1}, C_{2}, . . . . . . . . . . . C_{n}

are the Binomial coefficients in the expansion

(1 + x)^{n} .

‘n’ being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . . . (C_{0} + C_{1} + C_{2} + . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2}, . . . . . C_{r}

are binomial coefficients in the expansion of

{(1 + x)}^{n}

C_{1} - \frac{C_{2}}{2} + \frac{C_{3}}{3} - \frac{C_{4}}{4} + . . . . {(- 1)}^{n - 1} \frac{C_{n}}{n}

C_{0}, C_{1}, C_{2}, . . . . . . . . . . C_{n}

are the binomial coefficients in the expansion of

(1 + x)^{n}

. n being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2} . . . . . . . . . . ., C_{n}

are the binomial coefficient in the expansion of

(1 + x)^{n}, n

being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . . . . + C_{n - 1})

The value of $1. C_{1} + 3. C_{3} + 5. C_{5} + 7. C_{7} + . . . . w h e r e C_{0}, C_{1}, C_{2} . . . . . C_{n}$ are binomial coefficients in the expansion of $(1 + x)^{n}$ is:

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