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If (1+x)^n=C0+C1x+C2x^2++Cn x^n ,t h e n...

If `(1+x)^n=C_0+C_1x+C_2x^2++C_n x^n ,t h e nC_0C_2+C_1C_3+C_2C_4++C_(n-2)C_n=` `((2n)!)/((n !)^2)` b. `((2n)!)/((n-1)!(n+1)!)` c. `((2n)!)/((n-2)!(n+2)!)` d. none of these

A

`((2n)!)/((n!)^(2))`

B

`((2n)!)/((n-1)!(n+1)^(!))`

C

`((2n)!)/((n-2)!(n+2)!)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`(1+x)^(n)=C_(0)+.^(n)C_(1)x+.^(n)C_(2)x^(2)+.^(n)C_(3)x^(3)+"....."+C_(n-1)x^(n-1)+C_(n)x^(n) " "(1)`
`(x+1)^(n) = C_(0)x^(n)+C_(1)x^(n-1)+C_(2)x^(n-2)+"...."+C_(n-1)x+C_(n)" "(2)`
Multiplying Eqs. (1) and (2) and equating the coefficient of `x^(n-2)`. we get
`C_(0)C_(2)+C_(1)C_(3)+C_(2)C_(4)+"....."C_(n-2)C_(n)`
`=` coefficient of `x^(n-2)` in `(1+x)^(2n)`
`= .^(2n)C_(n-2)`
`= ((2n)!)/((n-2)!(n+2)!)`
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