定义: 1、从S中有序选取的r个元素称作S的一个r排列。S的不同r排列总数记作P（n，r），r=n时，称为S的全排列。
2、从S中无序选取的r个元素称作S的一个r组合。S的不同r组合总数记作C（n，r）。
推论 1、元素一次排成一个圆圈的排列称为环排列。S的环排列数等于 P（n，r）/r，其实就是线性排列数的1/r。
推论 2、C（n，r）= C（n-1，r-1）+C（n-1，r）。该公式就是杨辉三角形，也称作Pascal公式。
定义：设S=｛n1*a1，n2*a2，n3*a3，....，nk*ak｝为多重集，n=n1+n2+...+nk表示S中的元素总数。
（1）从S中有序选取的r个元素称为S的一个r排列。r=n的排列称为S的全排列。
（2）从S中无序选取的r个元素称为S的一个r组合。
定理：设S=｛n1*a1，n2*a2，n3*a3，....，nk*ak｝为多重集
（1）S的全排列数是n!/（n1! n2! n3!...nk!）.
（2）若r<=ni, i=1，2，3，...，k,那么S的 r 排列数是k^r。
（3）若r<=ni, i=1,2,3,..k，那么S的 r 组合数是C（k+r-1 , r）.即T={R*1, (K-1)**},等于（k+r-1）!/（r! *（k-1）!）.
格路径数：
定理：从（r，s）到 （p，q）的矩形格路径的条数等于二项式系数
C（p-r+q-s, p-r）=C（p-1+q-s, q-s）.
定理：令n为非负整数，则从（0，0）到（n，n）的下对角线矩形格路径的条数等于第n个Catalan数
Cn=1/(n+1) *C(2n，n).
定理：从（0，0）到（p，q）的下对角线矩形格路径的条数等于 （q-p+1）/(q+1)*C（p+q。q）。
前100个Catalan数：
“1”
“1”
"2", "5", "14", "42", "132", "429", "1430", "4862", "16796", "58786", "208012", "742900", "2674440", "9694845", "35357670", "129644790", "477638700", "1767263190", "6564120420", "24466267020", "91482563640", "343059613650", "1289904147324", "4861946401452", "18367353072152", "69533550916004", "263747951750360", "1002242216651368", "3814986502092304", "14544636039226909", "55534064877048198", "212336130412243110", "812944042149730764", "3116285494907301262", "11959798385860453492", "45950804324621742364", "176733862787006701400", "680425371729975800390", "2622127042276492108820", "10113918591637898134020", "39044429911904443959240", "150853479205085351660700", "583300119592996693088040", "2257117854077248073253720", "8740328711533173390046320", "33868773757191046886429490", "131327898242169365477991900", "509552245179617138054608572", "1978261657756160653623774456", "7684785670514316385230816156", "29869166945772625950142417512", "116157871455782434250553845880", "451959718027953471447609509424", "1759414616608818870992479875972", "6852456927844873497549658464312", "26700952856774851904245220912664", "104088460289122304033498318812080", "405944995127576985730643443367112", "1583850964596120042686772779038896", "6182127958584855650487080847216336", "24139737743045626825711458546273312", "94295850558771979787935384946380125", "368479169875816659479009042713546950", "1440418573150919668872489894243865350", "5632681584560312734993915705849145100", "22033725021956517463358552614056949950", "86218923998960285726185640663701108500", "337485502510215975556783793455058624700", "1321422108420282270489942177190229544600", "5175569924646105559418940193995065716350", "20276890389709399862928998568254641025700", "79463489365077377841208237632349268884500", "311496878311103321137536291518809134027240", "1221395654430378811828760722007962130791020", "4790408930363303911328386208394864461024520", "18793142726809884575211361279087545193250040", "73745243611532458459690151854647329239335600", "289450081175264899454283846029490767264392230", "1136359577947336271931632877004667456667613940", "4462290049988320482463241297506133183499654740", "17526585015616776834735140517915655636396234280", "68854441132780194707888052034668647142985206100", "270557451039395118028642463289168566420671280440", "1063353702922273835973036658043476458723103404520", "4180080073556524734514695828170907458428751314320", "16435314834665426797069144960762886143367590394940", "64633260585762914370496637486146181462681535261000", "254224158304000796523953440778841647086547372026600", "1000134600800354781929399250536541864362461089950800", "3935312233584004685417853572763349509774031680023800", "15487357822491889407128326963778343232013931127835600", "60960876535340415751462563580829648891969728907438000", "239993345518077005168915776623476723006280827488229600", "944973797977428207852605870454939596837230758234904050", "3721443204405954385563870541379246659709506697378694300", "14657929356129575437016877846657032761712954950899755100", "57743358069601357782187700608042856334020731624756611000", "227508830794229349661819540395688853956041682601541047340", "896519947090131496687170070074100632420837521538745909320"