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Given "^(8)C(1)x(1-x)^(7)+2*^(8)C(2)x^(2...

Given `"^(8)C_(1)x(1-x)^(7)+2*^(8)C_(2)x^(2)(1-x)^(6)+3*^(8)C_(3)x^(3)(1-x)^(5)+...+8*x^(8)=ax+b`, then `a+b` is

A

(a) `4`

B

(b) `6`

C

(c) `8`

D

(d) `10`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` `.^(n)C_(1)x(1-x)^(n-1)+2*^(n)C_(2)x^(2)(1-x)^(n-2)+3*^(n)C_(3)x^(3)(1-x)^(n-3)+....+n*^(n)C_(n)x^(n)`
`=sum_(r=1)^(n)r*^(n)C_(r )x^(r )(1-x)^(n-r)`
`=sum_(r=1)^(n)n^(n-1)C_(r-1)x^(r)(1-x)^(n-r)`
`=nsum_(r=1)^(n).^(n-1)C_(r-1)x*x^(r-1)(1-x)^((n-1)-(r-1))`
`=nxsum_(r=1)^(n).^(n-1)C_(r-1)x^(r-1)(1-x)^((n-1)-(r-1))`
`=nx[x+(1-x)]^(n-1)`
`=nx`
`.^(8)C_(1)x(1-x)^(7)+2*^(8)C_(2)x^(2)(1-x)^(6)+3*^(8)C_(3)x^(3)(1-x)^(5)+....+8.x^(8)`
`=8x`
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