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Prove that .^(n)C(0) - ^(n)C(1) + .^(n)C...

Prove that `.^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r )`.

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`.^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3)+"…"+(-1)^(r ) xx ,^(n)C_(r ) + "…"`
Coeffciint of `x^(r )` in
`(.^(n)C_(0) - .^(n)C_(1)x + .^(n)C_(2)x^(2) - .^(n)C_(3)x^(3) + "….." + (-1)^(r) xx .^(n)C_(r) + "…..") xx (1+x + x^(2)+x^(3)+"…."+x^(r ) + "....")`
`=` Coefficient of `x^(r)` in `(1-x)^(n)(1-x)^(-1)`
`=` Coefficient of `x^(r )` in `(1-x)^(n-1)`
`= (-1)^(r) xx .^(n-1)C_(r )`.
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