Solve 1-365!/365^40(365-40)!

$$1- \frac{ 365 ! }{ { 365 }^{ 40 } (365-40) ! }$$

Evaluate

$$\frac{19393328682731619082071437723120714139262612692853031533269435559474912532712558219274716441893}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}\approx 0.89123181$$

Solution Steps

The factorial of 365 is 25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{365^{40}\left(365-40\right)!}$$

Calculate 365 to the power of 40 and get 3102519917525622220066362032333600168597425643900453198929195853371717515992713742889463901519775390625.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{3102519917525622220066362032333600168597425643900453198929195853371717515992713742889463901519775390625\left(365-40\right)!}$$

Subtract 40 from 365 to get 325.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{3102519917525622220066362032333600168597425643900453198929195853371717515992713742889463901519775390625\times 325!}$$

The factorial of 325 is 74392417455820825437353449422105616638396717520380590731738414664170076047025418576647378675822937897054750306676289876155848679816329339933484229837744038579316146480420035168940232917224664376150369382451020933164230292622957073738126020152523218882320676726344979257687876563254167677401926546374562086915658699269826300119029558789735204443614385882152487400485317376300573220787727150660947912446615706931707574094166741527814381490079293895208733376861020373826490924313035525308093783171472696247763886024687394104351267644863982227540748624190929892712840958930646070438153721506685989683200000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{3102519917525622220066362032333600168597425643900453198929195853371717515992713742889463901519775390625\times 74392417455820825437353449422105616638396717520380590731738414664170076047025418576647378675822937897054750306676289876155848679816329339933484229837744038579316146480420035168940232917224664376150369382451020933164230292622957073738126020152523218882320676726344979257687876563254167677401926546374562086915658699269826300119029558789735204443614385882152487400485317376300573220787727150660947912446615706931707574094166741527814381490079293895208733376861020373826490924313035525308093783171472696247763886024687394104351267644863982227540748624190929892712840958930646070438153721506685989683200000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Multiply 3102519917525622220066362032333600168597425643900453198929195853371717515992713742889463901519775390625 and 74392417455820825437353449422105616638396717520380590731738414664170076047025418576647378675822937897054750306676289876155848679816329339933484229837744038579316146480420035168940232917224664376150369382451020933164230292622957073738126020152523218882320676726344979257687876563254167677401926546374562086915658699269826300119029558789735204443614385882152487400485317376300573220787727150660947912446615706931707574094166741527814381490079293895208733376861020373826490924313035525308093783171472696247763886024687394104351267644863982227540748624190929892712840958930646070438153721506685989683200000000000000000000000000000000000000000000000000000000000000000000000000000000 to get 230803956869564886122001438801926869917669235795861665714421981437596069771497019190096763226377797918885489054970417293911320549422312951008256435006859842507863436726710876096448128778355204455001644776419753714510093429122951677588012611623138734091748824428909629060120161706872491089264775821895339205425829406863683915740250661843101047267282932609979885465222943913403606060840163872309189951972496097217208956938261734879330309821355853022466572539683740448018369769809899331664213091818898712658111335884106580855750764320359735380530237931656453146845424888623080456068542482220818574954810094248065050109390547853532647116189649962504323558992248832000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{230803956869564886122001438801926869917669235795861665714421981437596069771497019190096763226377797918885489054970417293911320549422312951008256435006859842507863436726710876096448128778355204455001644776419753714510093429122951677588012611623138734091748824428909629060120161706872491089264775821895339205425829406863683915740250661843101047267282932609979885465222943913403606060840163872309189951972496097217208956938261734879330309821355853022466572539683740448018369769809899331664213091818898712658111335884106580855750764320359735380530237931656453146845424888623080456068542482220818574954810094248065050109390547853532647116189649962504323558992248832000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Reduce the fraction \frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{230803956869564886122001438801926869917669235795861665714421981437596069771497019190096763226377797918885489054970417293911320549422312951008256435006859842507863436726710876096448128778355204455001644776419753714510093429122951677588012611623138734091748824428909629060120161706872491089264775821895339205425829406863683915740250661843101047267282932609979885465222943913403606060840163872309189951972496097217208956938261734879330309821355853022466572539683740448018369769809899331664213091818898712658111335884106580855750764320359735380530237931656453146845424888623080456068542482220818574954810094248065050109390547853532647116189649962504323558992248832000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} to lowest terms by extracting and canceling out 10606731395068203626810159780886152372271406990210513912923641153289874123892296007998552037763817317353509321069080392498783112551937581670203806207334599250566559947403637826821556646401172850505814384607274468986306272190382551529068748967058974567205877736373405245725029275620223125879571558369810610048521650481830702946658511311817714696062207362103772617647320641542855166245125160152830464079302630089872368962644867444395409860890211824902807688497762670486980151318069518256818058928745130519700710312113632362534458082177872466036083299933472426109448027347533521761689886074195463372800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1-\frac{2366811012790553201133731599547824029530290453407681314014787412118503469866475155511660511232}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}$$

Convert 1 to fraction \frac{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}.

$$\frac{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}-\frac{2366811012790553201133731599547824029530290453407681314014787412118503469866475155511660511232}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}$$

Since \frac{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125} and \frac{2366811012790553201133731599547824029530290453407681314014787412118503469866475155511660511232}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125} have the same denominator, subtract them by subtracting their numerators.

$$\frac{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125-2366811012790553201133731599547824029530290453407681314014787412118503469866475155511660511232}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}$$

Subtract 2366811012790553201133731599547824029530290453407681314014787412118503469866475155511660511232 from 21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125 to get 19393328682731619082071437723120714139262612692853031533269435559474912532712558219274716441893.

$$\frac{19393328682731619082071437723120714139262612692853031533269435559474912532712558219274716441893}{21760139695522172283205169322668538168792903146260712847284222971593416002579033374786376953125}$$

Factor

$$\frac{19393328682731619082071437723120714139262612692853031533269435559474912532712558219274716441893}{5 ^ {31} \cdot 73 ^ {39}} = 0.8912318098179491$$