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Question Number 22739 by Tinkutara last updated on 22/Oct/17
If (1 + x)^n  = C_0  + C_1 x + C_2 x^2  + C_3 x^3   + ... + C_n x^n ,  Prove that ΣΣ_(0≤i<j≤n) (i + j)C_i C_j  =  n(2^(2n−1)  − (1/2)^(2n) C_n )
$${If}\:\left(\mathrm{1}\:+\:{x}\right)^{{n}} \:=\:{C}_{\mathrm{0}} \:+\:{C}_{\mathrm{1}} {x}\:+\:{C}_{\mathrm{2}} {x}^{\mathrm{2}} \:+\:{C}_{\mathrm{3}} {x}^{\mathrm{3}} \\ $$$$+\:…\:+\:{C}_{{n}} {x}^{{n}} , \\ $$$${Prove}\:{that}\:\underset{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} {\Sigma\Sigma}\left({i}\:+\:{j}\right){C}_{{i}} {C}_{{j}} \:= \\ $$$${n}\left(\mathrm{2}^{\mathrm{2}{n}−\mathrm{1}} \:−\:\frac{\mathrm{1}}{\mathrm{2}}\:^{\mathrm{2}{n}} {C}_{{n}} \right) \\ $$

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