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Stetemet - 1: sum(r=0)^(n) r. ""^(n)C(r)...

Stetemet - 1: `sum_(r=0)^(n) r. ""^(n)C_(r) = n 2^(n-1)`
Statement-2: ` sum_(r=0)^(n) r. ""^(n)C_(r) x^(r) = n (1 + x )^(n-1) x`

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
a
We have,
`sum_(r=0)^(n) r. ""^(n)C_(r) x^(r) = sum_(r=0)^(n) r (n)/(r) ""^(n-1)C_(r-1) x^(r-1) xxx`
`rArr sum_(r =0)^(n) r ""^(n)C_(r) x^(r) = nx sum_(r=1)^(n) ""^(n-1)C_(r-1) x^(r -1)`
`rArr sum_(r=0)^(n) r ""^(n)C_(r) x^(r) = nx (1 + x)^(n-1)` ...(i)
So, statement-2 is true.
Replacing x by in (i), we get `sum_(r=0)^(n) r ""^(n)C_(r) = n2^(n-1)`
So, statement -1 is also true stetement -2 is a correct
expansion for statement-1.
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