Question Number 209059 by hardmath last updated on 01/Jul/24
$$\mathrm{Compare}: \\ $$$$\mathrm{8}!\:\:\:\mathrm{and}\:\:\:\mathrm{8}!! \\ $$
Commented by mr W last updated on 01/Jul/24
$${if}\:\frac{{a}}{{b}}=\mathrm{1}\:\Rightarrow{a}={b} \\ $$$$\frac{\cancel{\mathrm{8}!}!}{\cancel{\mathrm{8}!}}=\mathrm{1}!=\mathrm{1}\:\Rightarrow\mathrm{8}!!=\mathrm{8}!\:\:\:\:\:\underset{\mid<>\mid} {\overset{−\vee\vee−} {\cup}} \\ $$
Commented by Frix last updated on 01/Jul/24
$${n}!!\neq\left({n}!\right)! \\ $$$${n}!!=\begin{cases}{\mathrm{2}^{{k}} {k}!,\:{n}=\mathrm{2}{k}}\\{\frac{\left(\mathrm{2}{k}\right)!}{\mathrm{2}^{{k}} {k}!},\:{n}=\mathrm{2}{k}−\mathrm{1}}\end{cases} \\ $$$$\Rightarrow\:\mathrm{8}!!=\left(\mathrm{2}×\mathrm{4}\right)!!=\mathrm{2}^{\mathrm{4}} ×\mathrm{4}!=\mathrm{16}×\mathrm{24}=\mathrm{384} \\ $$$$\mathrm{8}!=\mathrm{40320} \\ $$$$\mathrm{8}!=\mathrm{105}×\mathrm{8}!! \\ $$
Commented by A5T last updated on 01/Jul/24
$${It}\:{was}\:{probably}\:{a}\:{joke}. \\ $$
Commented by mokys last updated on 01/Jul/24
$${n}!\:=\:{n}\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)… \\ $$$${n}!!=\:{n}\:\left({n}−\mathrm{2}\right)\left({n}−\mathrm{4}\right)\left({n}−\mathrm{6}\right)… \\ $$$$ \\ $$$$\mathrm{8}!\:=\:\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1} \\ $$$$\mathrm{8}!!=\:\mathrm{8}×\mathrm{6}×\mathrm{4}×\mathrm{2} \\ $$