Question Number 23250 by Tinkutara last updated on 28/Oct/17
$${If}\:\left(\mathrm{1}\:+\:{x}\right)^{{n}} \:=\:{C}_{\mathrm{0}} \:+\:{C}_{\mathrm{1}} {x}\:+\:{C}_{\mathrm{2}} {x}^{\mathrm{2}} \:+\:{C}_{\mathrm{3}} {x}^{\mathrm{3}} \\ $$$$+\:…\:+\:{C}_{{n}} {x}^{{n}} ,\:{then}\:{prove}\:{that} \\ $$$$\underset{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} {\Sigma\Sigma}\left(\frac{{i}}{\:^{{n}} {C}_{{i}} }\:+\:\frac{{j}}{\:^{{n}} {C}_{{j}} }\right)\:=\:\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\left(\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\:^{{n}} {C}_{{r}} }\right). \\ $$