Question Number 217873 by ahmed2025 last updated on 22/Mar/25
$${What}\:{is}\:\mathrm{5}^{!\:} =\:? \\ $$
Commented by mr W last updated on 23/Mar/25
$${i}\:{don}'{t}\:{think}\:\mathrm{5}^{!} \:{is}\:{a}\:{common}\:{valid} \\ $$$${mathematical}\:{notation},\:{while}\:\mathrm{5}!, \\ $$$$\mathrm{5}!!\:{and}\:!\mathrm{5}\:{are}. \\ $$$${please}\:{show}\:{where}\:{you}\:{see}\:{it}\:{and} \\ $$$${in}\:{which}\:{context}. \\ $$
Answered by MrGaster last updated on 23/Mar/25
$$!\mathrm{5}=\mathrm{5}!\underset{{k}=\mathrm{0}} {\overset{\mathrm{5}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}!} \\ $$$$\mathrm{5}!=\mathrm{120} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{5}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}!}=\mathrm{1}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{24}}−\frac{\mathrm{1}}{\mathrm{120}}=\frac{\mathrm{44}}{\mathrm{120}} \\ $$$$\mathrm{5}!=\mathrm{120}×\frac{\mathrm{44}}{\mathrm{120}}=\mathrm{44} \\ $$