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Question Number 178486 by Acem last updated on 17/Oct/22

Reduce (((2n)!)/(1×3×5×...×(2n−1)))

$$\:{Reduce}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{1}×\mathrm{3}×\mathrm{5}×...×\left(\mathrm{2}{n}−\mathrm{1}\right)}\: \\ $$

Answered by Rasheed.Sindhi last updated on 17/Oct/22

Reduce (((2n)!)/(1×3×5×...×(2n−1))) =((1.2.3.4.5...(2n−1).2n)/(1.3.5.....(2n−1))) =2.4.6.....2n =2^n (1.2.3....n) =2^n n!

$$\:{Reduce}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{1}×\mathrm{3}×\mathrm{5}×...×\left(\mathrm{2}{n}−\mathrm{1}\right)}\: \\ $$$$=\frac{\cancel{\mathrm{1}}.\mathrm{2}.\cancel{\mathrm{3}}.\mathrm{4}.\cancel{\mathrm{5}}...\left(\cancel{\mathrm{2}{n}−\mathrm{1}}\right).\mathrm{2}{n}}{\cancel{\mathrm{1}}.\cancel{\mathrm{3}}.\cancel{\mathrm{5}}.....\left(\cancel{\mathrm{2}{n}−\mathrm{1}}\right)} \\ $$$$=\mathrm{2}.\mathrm{4}.\mathrm{6}.....\mathrm{2}{n} \\ $$$$=\mathrm{2}^{{n}} \left(\mathrm{1}.\mathrm{2}.\mathrm{3}....{n}\right) \\ $$$$=\mathrm{2}^{{n}} {n}! \\ $$

Commented by Acem last updated on 17/Oct/22

Good Sir!

$${Good}\:{Sir}!\: \\ $$

Commented by Acem last updated on 17/Oct/22

It prefer at 2nd line puttting the follow = ((2n.(2n−1)(2n−2)(2n−3)(2n−4)....6×5×4×3×2×1)/(1×3×5×....×(2n−1))) = 2n.(2n−2)(2n−4)..6×4×2 = 2^n .n(n−1)(n−2)...3×2×1= 2^n .n!

$${It}\:{prefer}\:{at}\:\mathrm{2}{nd}\:{line}\:{puttting}\:{the}\:{follow} \\ $$$$=\:\frac{\mathrm{2}{n}.\cancel{\left(\mathrm{2}{n}−\mathrm{1}\right)}\left(\mathrm{2}{n}−\mathrm{2}\right)\cancel{\left(\mathrm{2}{n}−\mathrm{3}\right)}\left(\mathrm{2}{n}−\mathrm{4}\right)....\mathrm{6}×\cancel{\mathrm{5}}×\mathrm{4}×\cancel{\mathrm{3}}×\mathrm{2}×\cancel{\mathrm{1}}}{\cancel{\mathrm{1}×\mathrm{3}×\mathrm{5}×....×\left(\mathrm{2}{n}−\mathrm{1}\right)}} \\ $$$$=\:\mathrm{2}{n}.\left(\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{4}\right)..\mathrm{6}×\mathrm{4}×\mathrm{2} \\ $$$$=\:\mathrm{2}^{{n}} .{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)...\mathrm{3}×\mathrm{2}×\mathrm{1}=\:\mathrm{2}^{{n}} .{n}! \\ $$

Commented by Rasheed.Sindhi last updated on 17/Oct/22

Ok sir, thank you!

$$\mathrm{Ok}\:\mathrm{sir},\:\mathrm{thank}\:\mathrm{you}! \\ $$

Commented by Acem last updated on 17/Oct/22

The best Rasheed, thank you

$${The}\:{best}\:{Rasheed},\:{thank}\:{you} \\ $$

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