whats-the-value-of-5-

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

$$=\mathrm{Subfactorial}5[\mathrm{No}.\mathrm{of}\mathrm{derangement}]$$$$$$$$=\lfloor \frac{5!}{\mathrm{e}}+\frac{1}{2}\rfloor$$$$=\lfloor 44.146+0.5\rfloor$$$$=\lfloor 44.646\rfloor$$$$=44$$

Commented by MathsFan last updated on 22/Oct/21

$$thankyousir$$$$butdoesthatmeanthequestionhas$$$$nouniquesolutionplease?$$

Commented by mr W last updated on 23/Oct/21

$$thereisuniquesolution!$$$$!5=44$$

Commented by mr W last updated on 23/Oct/21

Commented by MathsFan last updated on 25/Oct/21

$$thanksverymuch$$

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