Comparing $2013!$ and $1007^{2013}$

$1\times 2\times\ldots\times 2012\times 2013$

$=(1007-1006)\times(1007-1005)\times\ldots\times (1007+1005)\times(1007+1006)$

$=(1007^2-1006^2)\times (1007^2-1005^2)\times\ldots\times (1007^2-1^2)\times 1007$

$<(1007^2)^{1006}\times 1007$

$=1007^{2013}$

You can just take the inequality between the arithmetic and geometric means:

$$\sqrt[n]{a_1a_2\dots a_n}\leq \frac{a_1+a_2\dots+a_n}{n}$$

with $a_i=i$ and $n=2013$. Then $\sqrt[2013]{2013!} \leq 1007$.

Use Stirling’s approximation for the factorial:

$$n!\sim\sqrt{2\pi n}\left(\frac{n}e\right)^n\;,$$

so $$2013!\approx 112.46356(740.54132)^{2013}<1007^{2013}\;.$$

For a less computational approach, look at their ratio:

$$\begin{align*}\frac{2013!}{{1007}^{2013}}&=\frac{2013\cdot2012\cdot\ldots\cdot2\cdot1}{1007\cdot1007\cdot\ldots\cdot1007\cdot1007}\\\\ &=\underbrace{\frac{2013}{1007}\cdot\frac{2012}{1007}\cdot\frac{2011}{1007}\cdot\ldots\cdot\frac{1008}{1007}}_{1006\text{ factors}}\cdot\frac{1007}{1007}\cdot\underbrace{\frac{1006}{1007}\ldots\cdot\frac2{1007}\cdot\frac1{1007}}_{1006\text{ factors}}\;. \end{align*}$$

The $1006$ factors on the left are between $1$ and $2$; the factors on the right are between $1$ and $\frac1{1007}$. Try matching them up.

The number you used is still small enough to be calculated directly, so I used Python which have unlimited precision integer arithmetic to get the answer:

>>> math.factorial(2013) < 1007**2013
True

$2013!$ is a 5779-digit number:

28428672470596487143208829081843589752984259367654067435630178820900993877835804470129352637465623354790645508042117320630228341496254219920141445145336884286015878135691943249895619171517827978731216806669491194751589734007959706204777697708910258731579558923696248438260175260811845520119343741050973157820481207360858151329548873547155392262562914054345892038337798990005887015103189390432478646781349929343795332305445284870359141189456936265282903685914372066532795228775092127882107379575824697110714110216598705597141364745225997185014090908349189266792681168982706730982034513618993170855874333269695793623349197860089943442174193730066430857215683894230520597646816671734574453822639829410734166055686591247872808311907999627073361328419227145279682743490293656700038542243000003546564764348332370741823999560437942381028175926956969385684096585634140324840925655219566626681780413079707793203247656616899305549155300506512961224180922175695875855130161603169265027749450856546002014281973704117552075295088510263159717863893871565551141782499365149396562268214987359977682116903361743064161466851269709092940831385528741007252712667479425921896880812303322531002514033730272288019981274909925461757146500042810605575760186026914448743702765889614547298058019349143270586390379106418554798984347771317611058559518514307667291276804776127274419631816868303286813235147550700525598078555048080932058625744233064996833887297036248758245386006440826345817226225206354138940472165440197844386043987814485357207839608993158387923679839401916385370678987012450565846378062376136484680181696107424248833988290557328905530156622155731973183329522062412818891976040971797980881118048427449038178856677866729793522967701767979796809359702906622888622034875079794015190968478666786838663743343764094189633072364353610047074459411885945073259696421351331074156552963966241596119262109642041867876813009221657593868299350517580892902402234435384831115786326455758157066645451290187610438268934364523746868884697827904553630521630425186324509785150110425933785343763997021892027305674587963427362388597717089584800740389801871805661046797182251558328858758536549173591854185114387424505439992774084601227830850817225026918601823516287429424690520527151579756885433340958177062153568003130734463411930792098935999030660701955265631094948391159544743868355012643822204442366574187301773842976392960318817492317270290321216739518314342760193902063057941457483831918266124856180712708873479072562585163511439010136904713443685275196453218976196831564862256983178474297470717874357825058000841842254607086820958709964285390681344929611021650748906543867856832645308948315631325496978506390846757951539021919451562010507375333402971214499923228303735046388152853788197875547735758577143462711716276669975282702376254817177404843659528871571759852551928957855522749783350225715282936681198183637288250719167719456926810178261509011260991774619809583541100937880321896749918885701546906705975644873408883552724468281424925505085685052730630856891933179855224521270208562951496568791453972719099564639553188855510265294703433263335170190878675189768260844753417897774605623321349011902193140551488084053156588942044109445461823324089443588958669455041738856562689822933876705027721348392092785430536231155624184547493347003183551786284141683397325217471718551661094777193662269669826018290313483395680147568856167822253537316795890106429630267286486365176439534172233232992313064239259749985069822480881885861057794945255936109064568012376891509099628371428464518499556483198459740683706440681874790291565697629012736855369437572800033629881008632192695545890165897605607139837425754932756265266177004015134335575298431975102791240204949642576808809116443446034957239582791319783828624538693456582016675024102025726287573354996743008634325922561520602137325876607708685354396453575810753807003937278328177917617326892161755216237319977086984501681878924217092155504805061352726624769100848419300840110313542207006641815520942725354752516909996405280446089149387826652320289706871854037767039903259795085325094930026669064944815410134739486230574390677163549704597152234855003934633399961445040946853026906061000037823164158530258788716843440196064197752725288896695876633520564266421121601197005504235268082743668199089846809888151656764559897296502059421580244417539034627135829251486152815668187268736326174866477841051318182367428510319430606704509954580314723539663870622870718928895263796220304711503708642152686739807063295854270119924513338239613362449920358223131201084088421205205635444767996115430600182050994394325975798533130191543007073726382576912304230842839818060193895673180137969400931470686507837694128076839580668577787352528579884315928111505284991546171783935459962376675014391494104156202279442380953797629681894096893415368905476382033883187685926067674083948155038913063463236521164557031729533512019335884749555275632201290869633773423718239229618414281472158562815454469838680057650009657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and $1007^{2013}$ is a 6046-digit number:

12540770622340186642947856510793531886611912530777993058137911797578274638051861071232734697741620539036641265679212148809320334050510282436155553971915019593642601063230039285993210027316833229494019996552105028093917449384199390180627383327163664028628804352157574535269851978605241188192948561279832554499491936504664692559203076376719893471929908965648318021933036626374716898582548817753082949310563145935588658594122778469592203160059819913909820436184842607015273309981919874676707341507357662169232949073263880712042750658289370026048283334154877951521224477815038328455943600266175073512832249210166111988314930309304821959817581310558181836897479958903656656514762791074248152878686368693611099285066813477437771199117201326605447860462184325655714417886214306040804849382962033945680601679647853297347666822824453487097963119304626348810136207993650535865089187425317239955805145063258550512978367416014840538660072703157026590268795007278729829253999203060048785691953718047978785881698065677434716076521849039356546823668783369224020958737250660234554956739716809177968709319891768695573394838084371253037444242858895372290590446426196483469142832255798411673483749923961398830939535978830220266139409099479632602009340554505181898366032556799616519322340437892078308105009833157291531950868737961019457462715711059697605605584746734624571928423070253822044089781399853169426349766426960362396941448024944373806333452700555181556618725995616027828315510158543603768523621988960880542941087458463723237448681117793023018648287870342945384311867408054077292212210925380266103074112848724635330747161830392387189344531055031157090747598915495726074757461242709041546172692828724310644187606574049434803767037723948301935425349239802947032004919893998884555192641407250157544387478084907547282281953505422841458851462675189734371421060529653764746973347691680782684395887200638997846783924632124201532716043449511957770998671036230291812561515658633422643580811032376778997425322543070113242978519326057372277576195823313373629372025320377030489943394861333510103957725890977383320780109377420012231386881031913383420167855469002868871364135460841020314781548021485937450118030491172486762685200245901017203022100220814753330497186240660896740623396963257670973204205749212128070531423976841077346923345890195088924368145133802094133334537864233019513171379318911448439419992240669249787013765391668638784366386017011496054610213094995317870078469919339870711396773279210443985308668473738304238872501322875812630566563309665540439804488941791038761806235428421067411763142818094953690022795092210374864965969551738705254605247720560625049968655999533810963121595103986686326363266359249382233899303950092795146084513537407256446693882647520615279445154589341347556262892551088361851211811847715767738467590185362877136208892231689386725146948303729235800404475275414976627466325267089203567783645709897553753482273240202908417062634027964532930811188667938171439657369551726221747404444270311003296926405727557268176450104765828612273085156678023096572131405296866198347587390659767201963996330295350942534449264814668107804442445066512815976933049076518554357950313681286365958396312051603092688036238255084145979160688160078304417245389443869872897581719966860367139772722686460977719333699506892311676532183689726622250339552872134447245871511297460638151519876038608789297433386999083082138467728810780760166542249927744748937804148568614184557585685559281208539965195706721711352321923197643777718659709427260575996667182589184066512230516366684129134747081386608865809852630855464589822967021371423512066965790317995563448246932406250776727096599982867252055090048302831976569032105828395630242322603265927558194733033838615276769988605456876450633187485302834057397938810251134442357170056661712055027132368946638244916956094177953804741644846467185906553024396055755753715751904057531487694836417958996603739674280600406577887373711166328664912565111584138660091947985972717061773223068694903285516075689032920983166665990231408646216170879294842840440791918056727330600815113486979472534110271240954004872458447308011752579836210243314953217865316169220391629135927680054370364810412755363976249077729940161563021909329861737311243274443040402778189987110836924256648020285664514891319221306366363275630584541007747907990302696118647274087073517415065108868012582248621300278192836296644308531850281964753239278256454593247149123692878999651297025872315344133460680780715189588428373139640175330522626040622166563955858432908789411512263277119120066542480002599083434918179137387176236293954422279424266343400365508282860357065716664640191973915025253310996405274935381525732094178654969594130352052515861416045100888347159102227996288036710890962039178922782186371915310643849047358951146681533619733526718663319338810487251868787433028512549288496605489501829806509886788815137244575585003063804517480797026908414269969177462501408209498642124594908511942953262984079591669946781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(though I guess this is not really what you're looking for)

Let $j \in \{1,2,\dots,2013\}$.

When $j$ is small, $1007$ is much larger in size than $j$. [For e.g. $1007 = 1007 \cdot 1$] So for small $j$, the $j$-th factor in $1007^{2013}$ is much larger in size than the $j$-th factor in $2013!$.

But when $j$ is large, $j$ is not that much larger than $1007$. [For e.g. $2013 < 1007 \cdot 2$] So for large $j$, the $j$-th factor in $2013!$ is not that much larger in size than the $j$-th factor in $1007^{2013}$.

Does this line of reasoning help you find an answer to this question?

[Notice also that $1007$ is the median (midpoint) of $\{1,2,\dots,2013\}$.]