Question Number 164429 by malwan last updated on 17/Jan/22
$${If}\:{C}\left(\mathrm{2}{x}\:,\:{x}\right)\:=\:\mathrm{70} \\ $$$${prove}\:{that}\:{x}\:=\:\mathrm{4} \\ $$
Answered by alephzero last updated on 17/Jan/22
$${C}_{{x}} ^{\mathrm{2}{x}} \:=\:\mathrm{70} \\ $$$${C}_{{x}} ^{\mathrm{2}{x}} \:=\:{C}_{\mathrm{4}} ^{\mathrm{8}} \:=\:\frac{\mathrm{8}!}{\mathrm{4}!\left(\mathrm{8}−\mathrm{4}\right)!}\:=\:\frac{\mathrm{8}!}{\mathrm{4}!\mathrm{4}!}\:= \\ $$$$=\:\frac{\cancel{\mathrm{2}×\mathrm{3}×\mathrm{4}}×\mathrm{5}×\cancel{\mathrm{6}}×\mathrm{7}×\cancel{\mathrm{8}}}{\cancel{\left(\mathrm{2}×\mathrm{3}×\mathrm{4}\right)}\left(\mathrm{2}×\cancel{\mathrm{3}}×\cancel{\mathrm{4}}\right)}\:=\: \\ $$$$=\:\frac{\mathrm{5}×\mathrm{2}×\mathrm{7}×\cancel{\mathrm{2}}}{\cancel{\mathrm{2}}}\:=\:\mathrm{5}×\mathrm{2}×\mathrm{7}\:=\:\mathrm{10}×\mathrm{7}\:= \\ $$$$=\:\mathrm{70} \\ $$$$\Rightarrow\:{C}_{{x}} ^{\mathrm{2}{x}} \:=\:\mathrm{70}\:\Leftrightarrow\:{x}\:=\:\mathrm{4} \\ $$
Commented by malwan last updated on 17/Jan/22
$${direct}\:{proof}\:{please} \\ $$