Question Number 178486 by Acem last updated on 17/Oct/22
$$\:{Reduce}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{1}×\mathrm{3}×\mathrm{5}×…×\left(\mathrm{2}{n}−\mathrm{1}\right)}\: \\ $$
Answered by Rasheed.Sindhi last updated on 17/Oct/22
$$\:{Reduce}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{1}×\mathrm{3}×\mathrm{5}×…×\left(\mathrm{2}{n}−\mathrm{1}\right)}\: \\ $$$$=\frac{\cancel{\mathrm{1}}.\mathrm{2}.\cancel{\mathrm{3}}.\mathrm{4}.\cancel{\mathrm{5}}…\left(\cancel{\mathrm{2}{n}−\mathrm{1}}\right).\mathrm{2}{n}}{\cancel{\mathrm{1}}.\cancel{\mathrm{3}}.\cancel{\mathrm{5}}…..\left(\cancel{\mathrm{2}{n}−\mathrm{1}}\right)} \\ $$$$=\mathrm{2}.\mathrm{4}.\mathrm{6}…..\mathrm{2}{n} \\ $$$$=\mathrm{2}^{{n}} \left(\mathrm{1}.\mathrm{2}.\mathrm{3}….{n}\right) \\ $$$$=\mathrm{2}^{{n}} {n}! \\ $$
Commented by Acem last updated on 17/Oct/22
$${Good}\:{Sir}!\: \\ $$
Commented by Acem last updated on 17/Oct/22
$${It}\:{prefer}\:{at}\:\mathrm{2}{nd}\:{line}\:{puttting}\:{the}\:{follow} \\ $$$$=\:\frac{\mathrm{2}{n}.\cancel{\left(\mathrm{2}{n}−\mathrm{1}\right)}\left(\mathrm{2}{n}−\mathrm{2}\right)\cancel{\left(\mathrm{2}{n}−\mathrm{3}\right)}\left(\mathrm{2}{n}−\mathrm{4}\right)….\mathrm{6}×\cancel{\mathrm{5}}×\mathrm{4}×\cancel{\mathrm{3}}×\mathrm{2}×\cancel{\mathrm{1}}}{\cancel{\mathrm{1}×\mathrm{3}×\mathrm{5}×….×\left(\mathrm{2}{n}−\mathrm{1}\right)}} \\ $$$$=\:\mathrm{2}{n}.\left(\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{4}\right)..\mathrm{6}×\mathrm{4}×\mathrm{2} \\ $$$$=\:\mathrm{2}^{{n}} .{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)…\mathrm{3}×\mathrm{2}×\mathrm{1}=\:\mathrm{2}^{{n}} .{n}! \\ $$
Commented by Rasheed.Sindhi last updated on 17/Oct/22
$$\mathrm{Ok}\:\mathrm{sir},\:\mathrm{thank}\:\mathrm{you}! \\ $$
Commented by Acem last updated on 17/Oct/22
$${The}\:{best}\:{Rasheed},\:{thank}\:{you} \\ $$