Question Number 196815 by mathlove last updated on 01/Sep/23
$$!\mathrm{6}×\frac{\sqrt[{\mathrm{4}}]{!\mathrm{5}+\mathrm{9}!!!!!+\mathrm{7}!!!−\mathrm{16}}}{!\mathrm{10}}=? \\ $$
Answered by Tokugami last updated on 01/Sep/23
![9!!!!!=9×(9−5)=36 7!!!=7×(7−3)×(7−6)=28 !n=[((n!)/e)] !5=[((120)/e)]=44, !6=[((720)/e)]=265, !10=[((3628800)/e)]=1334961 265×(((44+36+28−16))^(1/4) /(1334961)) =((265((92))^(1/4) )/(1334961))≈0.00061478...](https://www.tinkutara.com/question/Q196824.png)
$$\mathrm{9}!!!!!=\mathrm{9}×\left(\mathrm{9}−\mathrm{5}\right)=\mathrm{36} \\ $$$$\mathrm{7}!!!=\mathrm{7}×\left(\mathrm{7}−\mathrm{3}\right)×\left(\mathrm{7}−\mathrm{6}\right)=\mathrm{28} \\ $$$$!{n}=\left[\frac{{n}!}{{e}}\right] \\ $$$$!\mathrm{5}=\left[\frac{\mathrm{120}}{{e}}\right]=\mathrm{44},\:!\mathrm{6}=\left[\frac{\mathrm{720}}{{e}}\right]=\mathrm{265},\:!\mathrm{10}=\left[\frac{\mathrm{3628800}}{{e}}\right]=\mathrm{1334961} \\ $$$$\mathrm{265}×\frac{\sqrt[{\mathrm{4}}]{\mathrm{44}+\mathrm{36}+\mathrm{28}−\mathrm{16}}}{\mathrm{1334961}} \\ $$$$=\frac{\mathrm{265}\sqrt[{\mathrm{4}}]{\mathrm{92}}}{\mathrm{1334961}}\approx\mathrm{0}.\mathrm{00061478}… \\ $$
Commented by mathlove last updated on 01/Sep/23
$${thanks} \\ $$
Commented by mokys last updated on 01/Sep/23
$${whats}\:{the}\:{formulla}\:{of}\:{n}!!!! \\ $$
Commented by Tokugami last updated on 01/Sep/23
$${n}!!!!!={n}\left({n}−\mathrm{5}\right)\left({n}−\mathrm{10}\right)\left({n}−\mathrm{15}\right)… \\ $$$$\mathrm{6}!!!!!=\mathrm{6}×\mathrm{1}=\mathrm{6} \\ $$$$\mathrm{7}!!!!!=\mathrm{7}×\mathrm{2}=\mathrm{14} \\ $$$$\mathrm{8}!!!!!=\mathrm{8}×\mathrm{3}=\mathrm{24} \\ $$$$ \\ $$