Question Number 157329 by MathsFan last updated on 22/Oct/21
$$\:\boldsymbol{{whats}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\: \\ $$$$\:\:\:\:!\mathrm{5} \\ $$
Answered by puissant last updated on 24/Oct/21
$$!\mathrm{5}=\mathrm{5}!\left(\frac{\mathrm{1}}{\mathrm{0}!}−\frac{\mathrm{1}}{\mathrm{1}!}+\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}−\frac{\mathrm{1}}{\mathrm{120}}\right)=\mathrm{5}!\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{24}}−\frac{\mathrm{1}}{\mathrm{120}}\right) \\ $$$$=\:\mathrm{5}!\left(\frac{\mathrm{60}}{\mathrm{120}}−\frac{\mathrm{20}}{\mathrm{120}}+\frac{\mathrm{5}}{\mathrm{120}}−\frac{\mathrm{1}}{\mathrm{120}}\right)=\:\mathrm{120}×\frac{\mathrm{44}}{\mathrm{120}}\:=\:\mathrm{44}.. \\ $$
Commented by MathsFan last updated on 22/Oct/21
$${thank}\:{you}\:{sir} \\ $$$${can}\:{this}\:{also}\:{be}\:{the}\:{answer}\:\left(\frac{\mathrm{5}!}{{e}}\right)=\mathrm{44}.\mathrm{146} \\ $$
Commented by puissant last updated on 22/Oct/21
$${yesss}\:\left(\frac{\mathrm{5}!}{{e}}\right)\:=\:\left(\frac{\mathrm{120}}{{e}}\right)\:=\:\mathrm{44},\mathrm{14} \\ $$
Commented by puissant last updated on 22/Oct/21
Commented by MathsFan last updated on 22/Oct/21
$${thank}\:{you}\:{senior} \\ $$
Commented by puissant last updated on 22/Oct/21
$${De}\:{rien}.. \\ $$
Commented by MathsFan last updated on 22/Oct/21
$${merci}\:{beaucoup} \\ $$😊
Answered by soumyasaha last updated on 22/Oct/21
$$\:\:\:=\:\mathrm{Subfactorial}\:\mathrm{5}\:\left[\mathrm{No}.\:\mathrm{of}\:\mathrm{derangement}\right]\: \\ $$$$\:\:\: \\ $$$$\:=\:\:\lfloor\frac{\mathrm{5}!}{\mathrm{e}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:\rfloor\:\:\:\: \\ $$$$\:\:\:\:\:\:=\:\:\lfloor\mathrm{44}.\mathrm{146}\:+\:\mathrm{0}.\mathrm{5}\:\rfloor\:\:\:\: \\ $$$$\:\:\:\:\:\:=\:\:\lfloor\mathrm{44}.\mathrm{646}\:\:\rfloor\:\:\:\: \\ $$$$\:\:\:\:\:\:=\:\:\mathrm{44}\:\:\: \\ $$
Commented by MathsFan last updated on 22/Oct/21
$${thank}\:{you}\:{sir} \\ $$$${but}\:{does}\:{that}\:{mean}\:{the}\:{question}\:{has} \\ $$$${no}\:{unique}\:{solution}\:{please}? \\ $$
Commented by mr W last updated on 23/Oct/21
$${there}\:{is}\:{unique}\:{solution}! \\ $$$$!\mathrm{5}=\mathrm{44} \\ $$
Commented by mr W last updated on 23/Oct/21
Commented by MathsFan last updated on 25/Oct/21
$${thanks}\:{very}\:{much} \\ $$