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Question Number 157329 by MathsFan last updated on 22/Oct/21

whats the value of !5

$$\:\boldsymbol{{whats}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\: \\ $$$$\:\:\:\:!\mathrm{5} \\ $$

Answered by puissant last updated on 24/Oct/21

!5=5!((1/(0!))−(1/(1!))+(1/(2!))−(1/(3!))+(1/(4!))−(1/(120)))=5!((1/2)−(1/6)+(1/(24))−(1/(120))) = 5!(((60)/(120))−((20)/(120))+(5/(120))−(1/(120)))= 120×((44)/(120)) = 44..

$$!\mathrm{5}=\mathrm{5}!\left(\frac{\mathrm{1}}{\mathrm{0}!}−\frac{\mathrm{1}}{\mathrm{1}!}+\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}−\frac{\mathrm{1}}{\mathrm{120}}\right)=\mathrm{5}!\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{24}}−\frac{\mathrm{1}}{\mathrm{120}}\right) \\ $$$$=\:\mathrm{5}!\left(\frac{\mathrm{60}}{\mathrm{120}}−\frac{\mathrm{20}}{\mathrm{120}}+\frac{\mathrm{5}}{\mathrm{120}}−\frac{\mathrm{1}}{\mathrm{120}}\right)=\:\mathrm{120}×\frac{\mathrm{44}}{\mathrm{120}}\:=\:\mathrm{44}.. \\ $$

Commented by MathsFan last updated on 22/Oct/21

thank you sir can this also be the answer (((5!)/e))=44.146

$${thank}\:{you}\:{sir} \\ $$$${can}\:{this}\:{also}\:{be}\:{the}\:{answer}\:\left(\frac{\mathrm{5}!}{{e}}\right)=\mathrm{44}.\mathrm{146} \\ $$

Commented by puissant last updated on 22/Oct/21

yesss (((5!)/e)) = (((120)/e)) = 44,14

$${yesss}\:\left(\frac{\mathrm{5}!}{{e}}\right)\:=\:\left(\frac{\mathrm{120}}{{e}}\right)\:=\:\mathrm{44},\mathrm{14} \\ $$

Commented by puissant last updated on 22/Oct/21

Commented by MathsFan last updated on 22/Oct/21

thank you senior

$${thank}\:{you}\:{senior} \\ $$

Commented by puissant last updated on 22/Oct/21

De rien..

$${De}\:{rien}.. \\ $$

Commented by MathsFan last updated on 22/Oct/21

merci beaucoup 😊

$${merci}\:{beaucoup} \\ $$😊

Answered by soumyasaha last updated on 22/Oct/21

= Subfactorial 5 [No. of derangement] = ⌊((5!)/e) + (1/2) ⌋ = ⌊44.146 + 0.5 ⌋ = ⌊44.646 ⌋ = 44

$$\:\:\:=\:\mathrm{Subfactorial}\:\mathrm{5}\:\left[\mathrm{No}.\:\mathrm{of}\:\mathrm{derangement}\right]\: \\ $$$$\:\:\: \\ $$$$\:=\:\:\lfloor\frac{\mathrm{5}!}{\mathrm{e}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:\rfloor\:\:\:\: \\ $$$$\:\:\:\:\:\:=\:\:\lfloor\mathrm{44}.\mathrm{146}\:+\:\mathrm{0}.\mathrm{5}\:\rfloor\:\:\:\: \\ $$$$\:\:\:\:\:\:=\:\:\lfloor\mathrm{44}.\mathrm{646}\:\:\rfloor\:\:\:\: \\ $$$$\:\:\:\:\:\:=\:\:\mathrm{44}\:\:\: \\ $$

Commented by MathsFan last updated on 22/Oct/21

thank you sir but does that mean the question has no unique solution please?

$${thank}\:{you}\:{sir} \\ $$$${but}\:{does}\:{that}\:{mean}\:{the}\:{question}\:{has} \\ $$$${no}\:{unique}\:{solution}\:{please}? \\ $$

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

there is unique solution! !5=44

$${there}\:{is}\:{unique}\:{solution}! \\ $$$$!\mathrm{5}=\mathrm{44} \\ $$

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

Commented by MathsFan last updated on 25/Oct/21

thanks very much

$${thanks}\:{very}\:{much} \\ $$

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