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Prove that .^n C0 . ^(2n) Cn- ^n C1 . ...

Prove that ` .^n C_0 . ^(2n) C_n- ^n C_1 . ^(2n-2)C_n+^n C_2 . ^(2n-4)C_n-=2^ndot`

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L.H.S. `= .^(n)C_(0).^(2n)C_(n)-.^(n)C_(1).^(2n-2)C_(n)+.^(n)C_(2).^(2n-4)C_(n)-"…."`
= Coefficient of `x^(n)` in `[.^(n)C_(0)(1+x)^(2n)- .^(n)C_(1)(1+x)^(2n-2)+.^(n)C_(2)(1+x)^(2n-4)-"….."]`
=Coefficient of `x^(n)` in `[(1+x)^(2) - 1]^(n)`
= Coefficient of `x^(n)` in `(2x+x^(2))^(n)`
= Coefficient of `x^(n)` in `x^(n)(2+x)^(n)`
`= 2^(n)`
= R.H.S.
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