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Prove that (^(2n)C0)^2+(^(2n)C1)^2+(^(2n...

Prove that `(^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot`

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`underset(r=0)overset(2n)sum(-1)^(r)(.^(2n)C_(r))^(2)=underset(r=0)overset(2n)sum(-1)^(r).^(2n)C_(r).^(2n)C_(r)`
`= underset(r=0)overset(2n)sum(-1)^(r) .^(2n)C_(r).^(2n)C_(2n-r)`
= Coefficient of `x^(2n)` in `(1-x)^(2n)(1+x)^(2n)`
= Coefficient of `x^(2n)` is `(1-x)^(2n)`
`= (-1)^(n).^(2n)C_(n)`
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