Solve 1-365!/365^24(365-24)!

$$1- \frac{ 365 ! }{ { 365 }^{ 24 } (365-24) ! }$$

Evaluate

$$\frac{14755228723361560701438151006321417951414372824732566737}{27408537393008102795780565710206448860593479156494140625}\approx 0.538344258$$

Solution Steps

The factorial of 365 is 25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{365^{24}\left(365-24\right)!}$$

Calculate 365 to the power of 24 and get 31262862963899867251437207763204230731614437162876129150390625.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{31262862963899867251437207763204230731614437162876129150390625\left(365-24\right)!}$$

Subtract 24 from 365 to get 341.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{31262862963899867251437207763204230731614437162876129150390625\times 341!}$$

The factorial of 341 is 1739394784987365478597195909819207036794753314616391322945878661135156557159810863161679503177535615565939695010521384572891622027344309905164623334044579443023088859101559737649225154583180358281474329651149746535612042678651389669462386663332191260467961608733633212413586911070774909418882810869676963472732478559563855903146367972596784528461229298553294661331368035359982666900925507637154491053743835801427587442788350736958758449420011899875146069094896529144291072265160584571766110362726970022770448949342780315886635095625670001870203438684873593945157119156027952294669064018627213007183777432809807258286977746575677823385600000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{31262862963899867251437207763204230731614437162876129150390625\times 1739394784987365478597195909819207036794753314616391322945878661135156557159810863161679503177535615565939695010521384572891622027344309905164623334044579443023088859101559737649225154583180358281474329651149746535612042678651389669462386663332191260467961608733633212413586911070774909418882810869676963472732478559563855903146367972596784528461229298553294661331368035359982666900925507637154491053743835801427587442788350736958758449420011899875146069094896529144291072265160584571766110362726970022770448949342780315886635095625670001870203438684873593945157119156027952294669064018627213007183777432809807258286977746575677823385600000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Multiply 31262862963899867251437207763204230731614437162876129150390625 and 1739394784987365478597195909819207036794753314616391322945878661135156557159810863161679503177535615565939695010521384572891622027344309905164623334044579443023088859101559737649225154583180358281474329651149746535612042678651389669462386663332191260467961608733633212413586911070774909418882810869676963472732478559563855903146367972596784528461229298553294661331368035359982666900925507637154491053743835801427587442788350736958758449420011899875146069094896529144291072265160584571766110362726970022770448949342780315886635095625670001870203438684873593945157119156027952294669064018627213007183777432809807258286977746575677823385600000000000000000000000000000000000000000000000000000000000000000000000000000000000 to get 54378460803182081048111838750972963629982336051514664303180648907174794130168860110757107599184796750517666895398493934885411490027133010940481651757719612725641486835359609370967974098620504260968764724753674819763831879573391018136311663182291788736182384778659813025495048611212055292463015834236549428235090587025710559418316041550219992046162963468688118267278069821176385868968475140798744622777518509428299226952184037248763089040394464288236629397838281001088486950945885770581556319074352353016783406179492715304677953963094552361799063509860046052002258138091075084793178863220119510497701683354872089143986619787513013038696255696249077223797463883824575912345600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{54378460803182081048111838750972963629982336051514664303180648907174794130168860110757107599184796750517666895398493934885411490027133010940481651757719612725641486835359609370967974098620504260968764724753674819763831879573391018136311663182291788736182384778659813025495048611212055292463015834236549428235090587025710559418316041550219992046162963468688118267278069821176385868968475140798744622777518509428299226952184037248763089040394464288236629397838281001088486950945885770581556319074352353016783406179492715304677953963094552361799063509860046052002258138091075084793178863220119510497701683354872089143986619787513013038696255696249077223797463883824575912345600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Reduce the fraction \frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{54378460803182081048111838750972963629982336051514664303180648907174794130168860110757107599184796750517666895398493934885411490027133010940481651757719612725641486835359609370967974098620504260968764724753674819763831879573391018136311663182291788736182384778659813025495048611212055292463015834236549428235090587025710559418316041550219992046162963468688118267278069821176385868968475140798744622777518509428299226952184037248763089040394464288236629397838281001088486950945885770581556319074352353016783406179492715304677953963094552361799063509860046052002258138091075084793178863220119510497701683354872089143986619787513013038696255696249077223797463883824575912345600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} to lowest terms by extracting and canceling out 1983997176626213749024926584637533026344015499484321352735142847857287948010409265793790683311876561504899964621375954278454506374939603485578398490394598427198210729912716575756147441946440096164806657258342679642182486180336741341730534787863280656471268709961800382909247570440102631055913206148225286461085483357002523139526325968743207352776089668662351723081091665332480229433868157148629341358176562711003341926930462559343583856369701073295088485061366353555207004302448791777170719632485450182222543332844108797808193155948029845883200797249933943093694839037344383086106901146246664836318996134298686403983583992187882517299200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{12653308669646542094342414703885030909179106331761573888}{27408537393008102795780565710206448860593479156494140625}$$

Convert 1 to fraction \frac{27408537393008102795780565710206448860593479156494140625}{27408537393008102795780565710206448860593479156494140625}.

$$\frac{27408537393008102795780565710206448860593479156494140625}{27408537393008102795780565710206448860593479156494140625}-\frac{12653308669646542094342414703885030909179106331761573888}{27408537393008102795780565710206448860593479156494140625}$$

Since \frac{27408537393008102795780565710206448860593479156494140625}{27408537393008102795780565710206448860593479156494140625} and \frac{12653308669646542094342414703885030909179106331761573888}{27408537393008102795780565710206448860593479156494140625} have the same denominator, subtract them by subtracting their numerators.

$$\frac{27408537393008102795780565710206448860593479156494140625-12653308669646542094342414703885030909179106331761573888}{27408537393008102795780565710206448860593479156494140625}$$

Subtract 12653308669646542094342414703885030909179106331761573888 from 27408537393008102795780565710206448860593479156494140625 to get 14755228723361560701438151006321417951414372824732566737.

$$\frac{14755228723361560701438151006321417951414372824732566737}{27408537393008102795780565710206448860593479156494140625}$$

Factor

$$\frac{14755228723361560701438151006321417951414372824732566737}{5 ^ {18} \cdot 73 ^ {23}} = 0.5383442579145289$$