Question Number 153006 by GalaxyBills last updated on 04/Sep/21
$${prove}\:{that}\: \\ $$$$\mathrm{4}!!=\mathrm{8} \\ $$$${please}\:{help} \\ $$
Commented by talminator2856791 last updated on 04/Sep/21
$$\: \\ $$$$\:\mathrm{that}\:\mathrm{is}\:\mathrm{wrong}. \\ $$$$\: \\ $$$$\:{n}!\:=\:{n}\centerdot\left({n}−\mathrm{1}\right)\centerdot\left({n}−\mathrm{2}\right)\centerdot…..\centerdot\mathrm{2}\centerdot\mathrm{1} \\ $$$$\: \\ $$$$\:{n}!!\:=\:{n}!\centerdot\left({n}!−\mathrm{1}\right)\centerdot\left({n}!−\mathrm{2}\right)\centerdot…..\centerdot \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left({n}!−{n}\right)\centerdot\left({n}!−{n}−\mathrm{1}\right)\centerdot…..\centerdot \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left({n}!−\mathrm{2}{n}\right)\centerdot\left({n}!−\mathrm{2}{n}−\mathrm{1}\right)\centerdot…..\centerdot \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:………………………………. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{n}\centerdot\left({n}−\mathrm{1}\right)\centerdot…..\centerdot\mathrm{2}\centerdot\mathrm{1} \\ $$$$\:\: \\ $$$$\:\mathrm{4}!!\:=\:\mathrm{4}!\centerdot\left(\mathrm{4}!−\mathrm{1}\right)\centerdot…..\centerdot\mathrm{2}\centerdot\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:=\mathrm{24}\centerdot\mathrm{23}\centerdot…..\centerdot\mathrm{2}\centerdot\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:=\:\mathrm{620}\:\mathrm{448}\:\mathrm{401}\:\mathrm{733}\:\mathrm{239}\:\mathrm{439}\:\mathrm{360}\:\mathrm{000} \\ $$$$\: \\ $$
Commented by mr W last updated on 04/Sep/21
$${n}!!\neq\left({n}!\right)! \\ $$$${n}!!={n}×\left({n}−\mathrm{2}\right)×\left({n}−\mathrm{4}\right)×…×\mathrm{2}\:{or}\:\mathrm{1} \\ $$
Commented by mr W last updated on 04/Sep/21
$${there}\:{is}\:{no}\:{why},\:{just}\:{know}\:{it}\:{if}\:{you} \\ $$$${didn}'{t}\:{already}\:{know}. \\ $$
Commented by talminator2856791 last updated on 04/Sep/21
$$\:\mathrm{why}? \\ $$
Commented by talminator2856791 last updated on 04/Sep/21
$$\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:{n}!!!? \\ $$
Commented by mr W last updated on 04/Sep/21
Commented by mr W last updated on 04/Sep/21
$${n}!!!={n}\left({n}−\mathrm{3}\right)\left({n}−\mathrm{6}\right)… \\ $$
Commented by Rasheed.Sindhi last updated on 04/Sep/21
$${If}\:{mod}\left({n},\mathrm{3}\right)=\mathrm{1}: \\ $$$${n}!!!={n}\left({n}−\mathrm{3}\right)\left({n}−\mathrm{6}\right)…×\mathrm{7}×\mathrm{4}×\mathrm{1} \\ $$$${If}\:\:{mod}\left({n},\mathrm{3}\right)=\mathrm{2}: \\ $$$${n}!!!={n}\left({n}−\mathrm{3}\right)\left({n}−\mathrm{6}\right)…×\mathrm{8}×\mathrm{5}×\mathrm{2} \\ $$$${If}\:{mod}\left({n},\mathrm{3}\right)=\mathrm{0}: \\ $$$${n}!!!={n}\left({n}−\mathrm{3}\right)\left({n}−\mathrm{6}\right)…×\mathrm{9}×\mathrm{6}×\mathrm{3} \\ $$
Answered by puissant last updated on 04/Sep/21
$${n}!!={n}\left({n}−\mathrm{2}\right)\left({n}−\mathrm{4}\right)….×\mathrm{4}×\mathrm{2} \\ $$$$\mathrm{4}!!=\mathrm{4}×\mathrm{2}=\mathrm{8}. \\ $$