Solve 15(285!/4!*281!)/frac{300!{5!*295!}}

$$\frac{ 15( \frac{ 285 ! }{ 4 ! \times 281 ! } ) }{ \frac{ 300 ! }{ 5 ! \times 295 ! } }$$

Evaluate

$$\frac{89715245}{435174168}\approx 0.206159399$$

Solution Steps

The factorial of 285 is 30443677370499005399989584609745133211935644628842033160341164810466069265398719498463862691230707022818960414124538337560160327022731991179747043070290883991335579280445867712773916495199386409310400266285370684871398634041776502820929860856713589496243719540759238436575784157203875968378438793208957089343428987445015659802542060017570668097888201519894401183823878767198397752095820301688938100218270188028144612485252509663566094062309107307224775530235575003917843346733017092345893009669707051837685760000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{15\times \frac{30443677370499005399989584609745133211935644628842033160341164810466069265398719498463862691230707022818960414124538337560160327022731991179747043070290883991335579280445867712773916495199386409310400266285370684871398634041776502820929860856713589496243719540759238436575784157203875968378438793208957089343428987445015659802542060017570668097888201519894401183823878767198397752095820301688938100218270188028144612485252509663566094062309107307224775530235575003917843346733017092345893009669707051837685760000000000000000000000000000000000000000000000000000000000000000000000}{4!\times 281!}}{\frac{300!}{5!\times 295!}}$$

The factorial of 4 is 24.

$$\frac{15\times \frac{30443677370499005399989584609745133211935644628842033160341164810466069265398719498463862691230707022818960414124538337560160327022731991179747043070290883991335579280445867712773916495199386409310400266285370684871398634041776502820929860856713589496243719540759238436575784157203875968378438793208957089343428987445015659802542060017570668097888201519894401183823878767198397752095820301688938100218270188028144612485252509663566094062309107307224775530235575003917843346733017092345893009669707051837685760000000000000000000000000000000000000000000000000000000000000000000000}{24\times 281!}}{\frac{300!}{5!\times 295!}}$$

The factorial of 281 is 4713010061646064073798405259537355169919319705617546004760389510795775949133827611161394075721143389537481186249731076232757989988635090077956440443918311268449597154809993507912159635869049780984268656886663199535033599616235249963018774453289155392623337120804217591846009894540943894265195333860197383606974889682754905349348817165535905064265678951450843967461203185329811151087566664346507617923407150285473997399781155375353011826831336578555476572669755631943824558272153357787416578368733184000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{15\times \frac{30443677370499005399989584609745133211935644628842033160341164810466069265398719498463862691230707022818960414124538337560160327022731991179747043070290883991335579280445867712773916495199386409310400266285370684871398634041776502820929860856713589496243719540759238436575784157203875968378438793208957089343428987445015659802542060017570668097888201519894401183823878767198397752095820301688938100218270188028144612485252509663566094062309107307224775530235575003917843346733017092345893009669707051837685760000000000000000000000000000000000000000000000000000000000000000000000}{24\times 4713010061646064073798405259537355169919319705617546004760389510795775949133827611161394075721143389537481186249731076232757989988635090077956440443918311268449597154809993507912159635869049780984268656886663199535033599616235249963018774453289155392623337120804217591846009894540943894265195333860197383606974889682754905349348817165535905064265678951450843967461203185329811151087566664346507617923407150285473997399781155375353011826831336578555476572669755631943824558272153357787416578368733184000000000000000000000000000000000000000000000000000000000000000000000}}{\frac{300!}{5!\times 295!}}$$

Multiply 24 and 4713010061646064073798405259537355169919319705617546004760389510795775949133827611161394075721143389537481186249731076232757989988635090077956440443918311268449597154809993507912159635869049780984268656886663199535033599616235249963018774453289155392623337120804217591846009894540943894265195333860197383606974889682754905349348817165535905064265678951450843967461203185329811151087566664346507617923407150285473997399781155375353011826831336578555476572669755631943824558272153357787416578368733184000000000000000000000000000000000000000000000000000000000000000000000 to get 113112241479505537771161726228896524078063672934821104114249348259098622779211862667873457817307441348899548469993545829586191759727242161870954570654039470442790331715439844189891831260857194743622447765279916788840806390789645999112450586878939729422960090899301222204304237468982653462364688012644737206567397352386117728384371611972861721542376294834820255219068876447915467626101599944316182830161771606851375937594747729008472283843952077885331437744074135166651789398531680586897997880849596416000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{15\times \frac{30443677370499005399989584609745133211935644628842033160341164810466069265398719498463862691230707022818960414124538337560160327022731991179747043070290883991335579280445867712773916495199386409310400266285370684871398634041776502820929860856713589496243719540759238436575784157203875968378438793208957089343428987445015659802542060017570668097888201519894401183823878767198397752095820301688938100218270188028144612485252509663566094062309107307224775530235575003917843346733017092345893009669707051837685760000000000000000000000000000000000000000000000000000000000000000000000}{113112241479505537771161726228896524078063672934821104114249348259098622779211862667873457817307441348899548469993545829586191759727242161870954570654039470442790331715439844189891831260857194743622447765279916788840806390789645999112450586878939729422960090899301222204304237468982653462364688012644737206567397352386117728384371611972861721542376294834820255219068876447915467626101599944316182830161771606851375937594747729008472283843952077885331437744074135166651789398531680586897997880849596416000000000000000000000000000000000000000000000000000000000000000000000}}{\frac{300!}{5!\times 295!}}$$

Divide 30443677370499005399989584609745133211935644628842033160341164810466069265398719498463862691230707022818960414124538337560160327022731991179747043070290883991335579280445867712773916495199386409310400266285370684871398634041776502820929860856713589496243719540759238436575784157203875968378438793208957089343428987445015659802542060017570668097888201519894401183823878767198397752095820301688938100218270188028144612485252509663566094062309107307224775530235575003917843346733017092345893009669707051837685760000000000000000000000000000000000000000000000000000000000000000000000 by 113112241479505537771161726228896524078063672934821104114249348259098622779211862667873457817307441348899548469993545829586191759727242161870954570654039470442790331715439844189891831260857194743622447765279916788840806390789645999112450586878939729422960090899301222204304237468982653462364688012644737206567397352386117728384371611972861721542376294834820255219068876447915467626101599944316182830161771606851375937594747729008472283843952077885331437744074135166651789398531680586897997880849596416000000000000000000000000000000000000000000000000000000000000000000000 to get 269145735.

$$\frac{15\times 269145735}{\frac{300!}{5!\times 295!}}$$

Multiply 15 and 269145735 to get 4037186025.

$$\frac{4037186025}{\frac{300!}{5!\times 295!}}$$

The factorial of 300 is 306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{4037186025}{\frac{306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000}{5!\times 295!}}$$

The factorial of 5 is 120.

$$\frac{4037186025}{\frac{306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000}{120\times 295!}}$$

The factorial of 295 is 130240536421542917278229579384676189809452723396833830881919418346829088259701652869791124822088676130693811076368225756785347739189353701535633041283562993445822106610934526642389209595125469386118872535411437545685252428896850271843922536464309018822342445871554626664317634424228848950164791490789924849157336730173758797906108816385979568498605532576905819795362719529920287521877108764591305813253343348372319239814648594495088292628894518916165874874657736591671523889732638401211216090244005499218918811963278481347594682368000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{4037186025}{\frac{306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000}{120\times 130240536421542917278229579384676189809452723396833830881919418346829088259701652869791124822088676130693811076368225756785347739189353701535633041283562993445822106610934526642389209595125469386118872535411437545685252428896850271843922536464309018822342445871554626664317634424228848950164791490789924849157336730173758797906108816385979568498605532576905819795362719529920287521877108764591305813253343348372319239814648594495088292628894518916165874874657736591671523889732638401211216090244005499218918811963278481347594682368000000000000000000000000000000000000000000000000000000000000000000000000}}$$

Multiply 120 and 130240536421542917278229579384676189809452723396833830881919418346829088259701652869791124822088676130693811076368225756785347739189353701535633041283562993445822106610934526642389209595125469386118872535411437545685252428896850271843922536464309018822342445871554626664317634424228848950164791490789924849157336730173758797906108816385979568498605532576905819795362719529920287521877108764591305813253343348372319239814648594495088292628894518916165874874657736591671523889732638401211216090244005499218918811963278481347594682368000000000000000000000000000000000000000000000000000000000000000000000000 to get 15628864370585150073387549526161142777134326807620059705830330201619490591164198344374934978650641135683257329164187090814241728702722444184275964954027559213498652793312143197086705151415056326334264704249372505482230291467622032621270704375717082258681093504586555199718116130907461874019774978894790981898880407620851055748733057966317548219832663909228698375443526343590434502625253051750956697590401201804678308777757831339410595115467342269939904984958928391000582866767916608145345930829280659906270257435593417761711361884160000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{4037186025}{\frac{306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000}{15628864370585150073387549526161142777134326807620059705830330201619490591164198344374934978650641135683257329164187090814241728702722444184275964954027559213498652793312143197086705151415056326334264704249372505482230291467622032621270704375717082258681093504586555199718116130907461874019774978894790981898880407620851055748733057966317548219832663909228698375443526343590434502625253051750956697590401201804678308777757831339410595115467342269939904984958928391000582866767916608145345930829280659906270257435593417761711361884160000000000000000000000000000000000000000000000000000000000000000000000000}}$$

Divide 306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000 by 15628864370585150073387549526161142777134326807620059705830330201619490591164198344374934978650641135683257329164187090814241728702722444184275964954027559213498652793312143197086705151415056326334264704249372505482230291467622032621270704375717082258681093504586555199718116130907461874019774978894790981898880407620851055748733057966317548219832663909228698375443526343590434502625253051750956697590401201804678308777757831339410595115467342269939904984958928391000582866767916608145345930829280659906270257435593417761711361884160000000000000000000000000000000000000000000000000000000000000000000000000 to get 19582837560.

$$\frac{4037186025}{19582837560}$$

Reduce the fraction \frac{4037186025}{19582837560} to lowest terms by extracting and canceling out 45.

$$\frac{89715245}{435174168}$$

Factor

$$\frac{5 \cdot 19 \cdot 47 \cdot 71 \cdot 283}{2 ^ {3} \cdot 3 \cdot 11 \cdot 13 \cdot 23 \cdot 37 \cdot 149} = 0.20615939914889433$$