Solve 200!/100!100!2^200

$$\frac { 200 ! } { 100 ! 100 ! 2 ^ { 200 } }$$

Evaluate

$$\frac{11318564332012910145675522134685520484313073709426667105165}{200867255532373784442745261542645325315275374222849104412672}\approx 0.056348479$$

Solution Steps

Multiply 100! and 100! to get \left(100!\right)^{2}.

$$\frac{200!}{\left(100!\right)^{2}\times 2^{200}}$$

The factorial of 200 is 788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000.

$$\frac{788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000}{\left(100!\right)^{2}\times 2^{200}}$$

The factorial of 100 is 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000.

$$\frac{788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000}{93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000^{2}\times 2^{200}}$$

Calculate 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000 to the power of 2 and get 8709782489089480079416590161944485865569720643940840134215932536243379996346583325877967096332754920644690380762219607476364289411435920190573960677507881394607489905331729758013432992987184764607375889434313483382966801515156280854162691766195737493173453603519594496000000000000000000000000000000000000000000000000.

$$\frac{788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000}{8709782489089480079416590161944485865569720643940840134215932536243379996346583325877967096332754920644690380762219607476364289411435920190573960677507881394607489905331729758013432992987184764607375889434313483382966801515156280854162691766195737493173453603519594496000000000000000000000000000000000000000000000000\times 2^{200}}$$

Calculate 2 to the power of 200 and get 1606938044258990275541962092341162602522202993782792835301376.

$$\frac{788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000}{8709782489089480079416590161944485865569720643940840134215932536243379996346583325877967096332754920644690380762219607476364289411435920190573960677507881394607489905331729758013432992987184764607375889434313483382966801515156280854162691766195737493173453603519594496000000000000000000000000000000000000000000000000\times 1606938044258990275541962092341162602522202993782792835301376}$$

Multiply 8709782489089480079416590161944485865569720643940840134215932536243379996346583325877967096332754920644690380762219607476364289411435920190573960677507881394607489905331729758013432992987184764607375889434313483382966801515156280854162691766195737493173453603519594496000000000000000000000000000000000000000000000000 and 1606938044258990275541962092341162602522202993782792835301376 to get 13996080838938649426554500942983193739421570235652683639204093286910404513060817589693846375521561189651176292387705957573821551681748140244715565857430844878200698301843889336575934561916587390308300073442341595072023994638965534355900229590218706268961044155442053655861958129277801745120015299070853647740238634788557470826496000000000000000000000000000000000000000000000000.

$$\frac{788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000}{13996080838938649426554500942983193739421570235652683639204093286910404513060817589693846375521561189651176292387705957573821551681748140244715565857430844878200698301843889336575934561916587390308300073442341595072023994638965534355900229590218706268961044155442053655861958129277801745120015299070853647740238634788557470826496000000000000000000000000000000000000000000000000}$$

Reduce the fraction \frac{788657867364790503552363213932185062295135977687173263294742533244359449963403342920304284011984623904177212138919638830257642790242637105061926624952829931113462857270763317237396988943922445621451664240254033291864131227428294853277524242407573903240321257405579568660226031904170324062351700858796178922222789623703897374720000000000000000000000000000000000000000000000000}{13996080838938649426554500942983193739421570235652683639204093286910404513060817589693846375521561189651176292387705957573821551681748140244715565857430844878200698301843889336575934561916587390308300073442341595072023994638965534355900229590218706268961044155442053655861958129277801745120015299070853647740238634788557470826496000000000000000000000000000000000000000000000000} to lowest terms by extracting and canceling out 69678259912715840635332721295555886924557765151526721073727460289947039970772666607023736770662039365157523046097756859810914315291487361524591685420063051156859919242653838064107463943897478116859007115474507867063734412121250246833301534129565899945387628828156755968000000000000000000000000000000000000000000000000.

$$\frac{11318564332012910145675522134685520484313073709426667105165}{200867255532373784442745261542645325315275374222849104412672}$$

Factor

$$\frac{3\times 5\times 11\times 13^{2}\times 17\times 37\times 53\times 59\times 61\times 101\times 103\times 107\times 109\times 113\times 127\times 131\times 137\times 139\times 149\times 151\times 157\times 163\times 167\times 173\times 179\times 181\times 191\times 193\times 197\times 199}{2^{197}}$$