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Prove that ^m C1^n Cm-^m C2^(2n)Cm+^m C3...

Prove that `^m C_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-=(-1)^(m-1)n^mdot`

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`.^(m)C_(1).^(n)C_(m)-.^(m)C_(2).^(2n)C_(3)+.^(m)C_(3).^(3n)C_(m)-"...."+(-1)^(m-1).^(m)C_(m).^(mn)C_(m)`
`= "Coefficient of" x^(m) " in"`
`.^(m)C_(1)(1+x)^(n)-.^(m)C_(2)(1+x)^(2n)+.^(m)C_(3)(1+x)^(3n)-"...."+(-1)^(m-1).^(m)C_(m)(1+x)^(mn)`
`=` Coefficient of `x^(m)` in
`.^(m)C_(0) - [.^(m)C_(0) - .^(m)C_(1)(1+x)^(n)+.^(m)C_(2)(1+x)^(2n)-"...."+(-1)^(m).^(m)C_(m)(1+x)^(mn)]`
`=` Coefficient of `x^(m)` in `[1-{1-(1+x)^(n)}^(m)]`
`=` Coefficient of `x^(m)` in `[1-{-nx-.^(n)C_(2)x^(2)-.^(n)C_(3)x^(3)-"......"-.^(n)C_(n)x^(n)}^(m)]`
`= - (-n)^(m)`
`= (-1)^(m-1)n^(m)`
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