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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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