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Question Number 63561 by aliesam last updated on 05/Jul/19

Answered by MJS last updated on 05/Jul/19

=Π_(k=1) ^n (2k−1)×Π_(k=1) ^n 2k=  =[1×3×5×...×(2n−1)]×[2×4×6×...×2n]=  =1×2×3×4×...×(2n−1)(2n)=  =(2n)!

$$=\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{2}{k}−\mathrm{1}\right)×\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\mathrm{2}{k}= \\ $$$$=\left[\mathrm{1}×\mathrm{3}×\mathrm{5}×...×\left(\mathrm{2}{n}−\mathrm{1}\right)\right]×\left[\mathrm{2}×\mathrm{4}×\mathrm{6}×...×\mathrm{2}{n}\right]= \\ $$$$=\mathrm{1}×\mathrm{2}×\mathrm{3}×\mathrm{4}×...×\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}\right)= \\ $$$$=\left(\mathrm{2}{n}\right)! \\ $$

Commented by aliesam last updated on 05/Jul/19

god bless you sir

$${god}\:{bless}\:{you}\:{sir} \\ $$

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