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Prove that .^n C0 .^n C0-^(n+1)C1 . ^n C...

Prove that `.^n C_0 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2- .. =(-1)^n`.

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`.^(n)C_(0).^(n)C_(0)-.^(n)C_(1).^(n+1)C_(1)+.^(n)C_(2).^(n+2)C_(2)-"....."`
`= .^(n)C_(0).^(n)C_(n)-.^(n)C_(1).^(n+1)C_(n)+.^(n)C_(2).^(n+2)C_(n)-"...."`
= Coefficient of `x^(n)` in `[.^(n)C_(0)(1+x)^(n)-.^(n)C_(1)(1+x)^(n+1)+.^(n)C_(2)(1+x)^(n+2)+"....."+(-1)^(n).^(n)C_(n)(1+x)^(2n)]`
= Coefficient of `x^(2)` in `(1+x)^(n)[.^(n)C_(0) - .^(n)C_(1)(1+x)+.^(n)C_(2)(1+x)^(2)-"......"+(-1)^(n).^(n)C_(n)(1+x)^(n)]`
= Coefficient of `x^(n)` in `(1+x)^(n)[1-(1+x)]^(n)`
= Coefficient of `x^(n)` in `(1+x)^(n) (-x)^(n)`
`= (-1)^(n)`
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