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Prove that ""^(10)C(2)+2xx^(10)C(3)+^(10...

Prove that `""^(10)C_(2)+2xx^(10)C_(3)+^(10)C_(4)=^(12)C_(4)`

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`S = .^(10)C_(1)(x-1)^(2).^(10)C_(2)(x-2)^(2)+.^(10)C_(3)(x-3)^(2)+"...."-.^(10)C_(10)(x-10)^(10)`
`= underset(r=1)overset(10)sum(-1)^(r+1).^(10)C_(r)(x-r)^(2)`
`= underset(r=1)overset(10)(-1)^(r+1).^(10)C_(r)(x^(2) - 2xr+r^(2))`
`= underset(r=1)overset(10)sum(-1)^(r+1)C_(r)(x^(2)) - 2x underset(r=1)overset(10)sum(-1)^(r+1).^(10)C_(r)r + underset(r=1)overset(10)sum(-1)^(r+1).^(10)C_(r)r^(2)`
`= x^(2) underset(r=1)overset(10)sum (-1)^(r+1) .^(10)C_(r) - 2x(0) + 0`
`=x^(2)(.^(10)C_(1) - .^(10)C_(2) + .^(10)C_(3)-.^(10)C_(4)+"...."-.^(10)C_(10))`
`= x^(2)(1) = x^(2)`
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