Solve 365!/365^26*(365-26)!

$$\frac{ 365 ! }{ { 365 }^{ 26 } \times (365-26) ! }$$

Evaluate

$$\frac{293404921431764018083611912153686096722045117620887375314944}{730300478836700898993573173348450829890513252124786376953125}\approx 0.40175918$$

Solution Steps

The factorial of 365 is 25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{365^{26}\left(365-26\right)!}$$

Calculate 365 to the power of 26 and get 4164994918365559814572722004252883639219333391024172306060791015625.

$$\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{4164994918365559814572722004252883639219333391024172306060791015625\left(365-26\right)!}$$

Subtract 26 from 365 to get 339.

$$\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{4164994918365559814572722004252883639219333391024172306060791015625\times 339!}$$

The factorial of 339 is 15002542565010914943912333188021451067748432936142757658667230128817979620146721262391577567513676173589267681650175820018040555695569345395589299068868202889624709842173190768063008060920996707620099444981453739310091794709775656973110114398242118858616194658734114304067508289380497752448532093062592405319410717263790373496173606801766297468183795916450704341309022212868575702095269170580942651834947695371981951378198643582531985936001482662369726316154015259136545387831297089630551236525159306734262971790087806761140547659355442486374016203940603708341876135553113268023711092104771545689009638026650053978669809785886474240000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{4164994918365559814572722004252883639219333391024172306060791015625\times 15002542565010914943912333188021451067748432936142757658667230128817979620146721262391577567513676173589267681650175820018040555695569345395589299068868202889624709842173190768063008060920996707620099444981453739310091794709775656973110114398242118858616194658734114304067508289380497752448532093062592405319410717263790373496173606801766297468183795916450704341309022212868575702095269170580942651834947695371981951378198643582531985936001482662369726316154015259136545387831297089630551236525159306734262971790087806761140547659355442486374016203940603708341876135553113268023711092104771545689009638026650053978669809785886474240000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Multiply 4164994918365559814572722004252883639219333391024172306060791015625 and 15002542565010914943912333188021451067748432936142757658667230128817979620146721262391577567513676173589267681650175820018040555695569345395589299068868202889624709842173190768063008060920996707620099444981453739310091794709775656973110114398242118858616194658734114304067508289380497752448532093062592405319410717263790373496173606801766297468183795916450704341309022212868575702095269170580942651834947695371981951378198643582531985936001482662369726316154015259136545387831297089630551236525159306734262971790087806761140547659355442486374016203940603708341876135553113268023711092104771545689009638026650053978669809785886474240000000000000000000000000000000000000000000000000000000000000000000000000000000000 to get 62485513545833472034109881987220744174610977406098336655090925915631895359597605556801929100408785122371193480588790361179135292037819522016091668582216624162270028321897394846016977309718101433220318099493775468889395395516344819658531320747462683753475057893194355617746919537939676266503236885597415021361220833676904340853071887489460569607987414249749651165759193133743522575412498711686326999909736962382138731332626516840317944949168125796104234531024710939882815887871016316937449030607905746340011810317948223188422635990454301693985511782828226973244789032665029137239703761018633963999105630196246628223284607738411429722962814086016761369160058012097025409024000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{62485513545833472034109881987220744174610977406098336655090925915631895359597605556801929100408785122371193480588790361179135292037819522016091668582216624162270028321897394846016977309718101433220318099493775468889395395516344819658531320747462683753475057893194355617746919537939676266503236885597415021361220833676904340853071887489460569607987414249749651165759193133743522575412498711686326999909736962382138731332626516840317944949168125796104234531024710939882815887871016316937449030607905746340011810317948223188422635990454301693985511782828226973244789032665029137239703761018633963999105630196246628223284607738411429722962814086016761369160058012097025409024000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Reduce the fraction \frac{25104128675558732292929443748812027705165520269876079766872595193901106138220937419666018009000254169376172314360982328660708071123369979853445367910653872383599704355532740937678091491429440864316046925074510134847025546014098005907965541041195496105311886173373435145517193282760847755882291690213539123479186274701519396808504940722607033001246328398800550487427999876690416973437861078185344667966871511049653888130136836199010529180056125844549488648617682915826347564148990984138067809999604687488146734837340699359838791124995957584538873616661533093253551256845056046388738129702951381151861413688922986510005440943943014699244112555755279140760492764253740250410391056421979003289600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{62485513545833472034109881987220744174610977406098336655090925915631895359597605556801929100408785122371193480588790361179135292037819522016091668582216624162270028321897394846016977309718101433220318099493775468889395395516344819658531320747462683753475057893194355617746919537939676266503236885597415021361220833676904340853071887489460569607987414249749651165759193133743522575412498711686326999909736962382138731332626516840317944949168125796104234531024710939882815887871016316937449030607905746340011810317948223188422635990454301693985511782828226973244789032665029137239703761018633963999105630196246628223284607738411429722962814086016761369160058012097025409024000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} to lowest terms by extracting and canceling out 85561375566077874289500025212934838120752781588939164772086546828415040021149269699576965814726434427501292246911158973540387544201293922959220221252138969604890923318643978599109342847440059348145879647159853357002867266704189293674768621177474584115545485163092995640385008212873151244433034593247597311587264246895054473845365101291323415248235711086007923196528017307766095801012081988469438561246186075168334566453789139181627732291258455808827345396815868274763110414975366214299237520807549171218843510990344522934629685869761507930101811163098755524137262335576349106697727322160025221507633091870738589097101258935133798400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{293404921431764018083611912153686096722045117620887375314944}{730300478836700898993573173348450829890513252124786376953125}$$

Factor

$$\frac{2 ^ {25} \cdot 3 ^ {12} \cdot 7 ^ {6} \cdot 11 ^ {4} \cdot 13 ^ {2} \cdot 17 ^ {2} \cdot 19 ^ {3} \cdot 23 \cdot 29 \cdot 31 \cdot 43 \cdot 59 \cdot 71 \cdot 89 \cdot 173 \cdot 179 \cdot 181 \cdot 347 \cdot 349 \cdot 353 \cdot 359}{5 ^ {19} \cdot 73 ^ {25}} = 0.401759179864061$$