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The value of the expansion (sumsum)(0 le...

The value of the expansion `(sumsum)_(0 le i lt j le n) (-1)^(i+j-1)"^(n)C_(i)*^(n)C_(j)=`

A

`"^(2n-1)C_(n)`

B

`"^(2n)C_(n)`

C

`"^(2n+1)C_(n)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` Let the required value be `S`
`sum_(j=0)^(n)sum_(i=0)^(n)(-1)^(i+j-1)'^(n)C_(i)*^(n)C_(j)=sum_(i=0)^(n)(-1)^(2i-1)('^(n)C_(i))^(2)+2S`
`:.0=-sum_(i=0)^(n)('^(n)C_(i))^(2)+2S`
`:.2S=^(2n)C_(n)`
`:.S=^(2n-1)C_(n-1)=^(2n-1)C_(n)`
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