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The coefficient of x^(n-2) in the polyno...

The coefficient of `x^(n-2)` in the polynomial `(x-1)(x-2)(x-3)...(x-n)` is

A

`(n(n^(2)+2)(3n+1))/(24)`

B

`(n(n^(2)-1)(3n+2))/(24)`

C

`(n(n^(2)+1)(3n+4))/(24)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let `f(x)=(x-1)(x-2)(x-3)"…."(x-n)`
`=x^(n)-S_(1)x^(n-1)+S_(2)x^(n-2)-"..."+(-1)^(n)(1*2*3*"..."n)`
So, coefficient of `x^(n-2)` in `f(x)=S_(2)=(1*2*3*"..."n)`
=Sum of product of first n natural number taken 2 at time
`=(1)/(2)[(1+2+"......"+n)^(2)-(1^(2)+2^(2)+"....."+n^(2))]`
`=(1)/(2)[{(n(n+1))/(2)}^(2)-(n(n+1)(2n+1))/(6)]`
`=(1)/(2)*(n(n+1))/(2)[(n(n+1))/(2)-(2n+1)/(3)]`
`=(1)/(2)*(n(n+1))/(2)[(3n^(2)+3n-4n-2)/(6)]`
`=(n(n+1)(3n^(2)+3n-4n-2))/(24)=(n(n+1)(3n+2)(n-1))/(24)`
`=(n(n^(2)-1)(3n+2))/(24)`.
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