组合数学期末考试复习资料

111)1000012)1000001)1000014,0000.999916560.199996560=3439:410,194,14:10494=34392)010192923934940100009+92+93+94=9(194)/(19)=738001000099997380=26192200000900000}9,7,5,3,1{,}8,7,6,5,4,3,2{∈∈fa}7,5,3{∈a48481243pp=⋅⋅}8,6,4,2{∈a48482054pp=⋅⋅2000009000004832p31400mmm1400krkrrppppppmk,,,,212121LL⋅⋅⋅=kirspppiiskssk,1,0,2121=≤≤⋅⋅⋅L)1()1()1(21+⋅⋅+⋅+krrrL2140031211124453Q8873Q66P33533Q555434!5435nn3n3n3nn3nn3nn3nCn3n4n4nn4nn4nn4nCn41237521400⋅⋅=36130033313003A={i|i1(mod3)}={1,4,7,…,298},B={i|i2(mod3)}={2,5,8,…,299},C={i|i3(mod3)}={3,6,9,…,300}.1)3A2)3B3)3C4)A,B,C3C(100,3)+1003=485100+1000000=1485100{124n}4414n4A{i|i0(mod4)}={1,5,9,…,4n},B={i|i1(mod4)}={1,5,9,…,4n-3},C={i|i2(mod4)}={2,6,10,…,4n-2},D={i|i3(mod4)}={3,7,11,…,4n-1}.4A4B4C4D1A2B,1C1A1C2D2A1B1D2A2C1B2C1D2B2D4C(n,4)+4C(n,2)[C(n,1)]2+2[C(n,2)]276900011001010100“0”“1”145!14xx·5!=14!x=14!/5!=726485760141C(14,5)9C(14,5)·9!=7264857609191621112732891446·7·8·9·10·11·12·13·14=7264857605951.Q101092.Q55254!25101552(1)525(2)525(4)(3)5255(1)5252411234113CCCC⋅⋅⋅(2)525141112424213CCCCC⋅⋅⋅⋅(3)5251411244113CCCC⋅⋅⋅112861210000713nabmm=a+baC(n,m)mm-1∑=⋅-nmmnCm2),()1(141100000010000000999999d1d2d3d4d5d6dii=1,2,…,6091066610600000099999916106/10610511000000161051600001000000010000000d1d2d3d4d5d6059490390029000190000061055949039002900019000048889515nnnn,22n!n)2n!n)/(2n)n!n116mkrkrrpppm⋅⋅⋅=L2121kppp,,,21Lmt=(r1+1)·(r2+1)·…·(rk+1)8mr1r2…rktk17nn3CnCnnnnn3Cnnn3Cnn3CnCnnn3CnnCnnCCn418nrn1,2,…,nra1,a2,…,ar.1a1a2…arn,bi=ai+i-1,i=1,2,…,r.b1,b2,…,br1b1b2…brn+r-1b1,b2,…,br1,,2,…,n+r-1r1,2,…,nnr1,,2,…,n+r-1n+r-1r1,,2,…,n+r-1n+r-1rb1,b2,…,br.1b1b2…brn+r-1,a1,a2,…,ar1,,2,…,nr1,2,…,n+r-1n+r-1rrrnc1-+91,,2,…,nnrnrnr1r19{1,2,…,9}7756A{1,2,…,9}77B{1,2,…,9}7756|A||B||A|=P(9,7)|B|=2C(7,5)P(6,6)|A||B|=P(9,7)2C(7,5)P(6,6)20{1,2,…,(n+1)}3(x,y,z)zxzy1z=k+1(x,y,z)k22xyx=yxyC(n1,3)C(n1,2)C(n1,3)3)2,1()3,1(212+++=∑=nCnCknk1zxzyz=k+1xy{1,2,…,k}C(k,1)C(k,1)=k22zxzyxy{1,2,…,(n+1)}3(x,y,z)zxyC(n1,3)xy{1,2,…,(n+1)}2(x,y,z)zxyC(n1,2)xy{1,2,…,(n+1)}3(x,y,z)zyxC(n1,3)31z=k+1(x,y,z)k2k1,2,…,n{1,2,…,(n+1)}3(x,y,z)∑==+++nkkn1222221L102{1,2,…,(n+1)}3(x,y,z)zxzyxyxyx=yxy2C(n1,3)C(n1,2))2,1()3,1(212+++=∑=nCnCknk211,,2,…,nrra1,a2,…,ar,1a1a2…arn,1a1-1a2-2…ar-r+1n-r+1,bi=ai-i+1,i=1,2,…,r.b1,b2,…,br,1b1b2…brn-r+1b1,b2,…,br1,,2,…,n-r+1r1,2,…,nnr1,,2,…,n-r+1n-r+1r1,,2,…,n-r+1n-r+1rb1,b2,…,br,1b1b2…brn-r+1,ai=bi+i-1,i=1,2,…,r.a1,a2,…,ar1a1a2…arna1,a2,…,ar1,,2,…,nr1,2,…,nr1nr1r1,,2,…,nnr1nnr1nr1nr1r1n22n1,0,!1≥≤≤⋅=∑≥iiaianiiirrnc1+-21)2()4(1-+---+--rrnrrncc111nn=0100·111·1nn=111·1nknk11,0,!,,0,1≥≤≤⋅=∃≤∀∑=iiaiamunmmiuii01,0,!1,1≠≥≤≤⋅=-∃∑=sisiiaiiaians!!)!1(1!!1ssssianssii+⋅=+≤+⋅=∑=!!0sssn⋅≤-1,0,!!1≥≤≤⋅=-∑=iiaiasnisii1,0,!!1≥≤≤+⋅=∑=iiasianisii1,0,!11!,11≥≤≤⋅⋅==∑=iiasiansaisiis1,0,!)1(!,11≥≤≤⋅+⋅=≠∑=iiasaiansaissiis1,0,!1≥≤≤⋅=∑≥iiaianniii,,2,1,0,!1tiiaiakitiiL=≤≤⋅=∑=,,2,1,0,1!11tiiaiakitiiL=≤≤+⋅=+∑=,)!1(11!1,1+⋅=+⋅=+≡∑=tiikiatiiiaiaiaiaiikiaiiiaitiiiitiiiiitii≤≤⋅+⋅+=+⋅+⋅=+=≠∃∑∑∑+==-=≤≤0,!!)1(1!!1},{min,1011100000122)!1(+jjjba=23npn1nn=111·p0nk0,,0,!!00≥≤≤⋅=⋅=∑∑≥≥iibaibianiiiiii}{max0iiibaij≠=≥iibai≠∃,jjba0,,0,!!00≥≤≤⋅=⋅∑∑==iibaibiaiijiijii∑∑-=-=⋅-⋅-1010!)(!)(jiiijiiiiabiab}{min0iiibaij≠=≥jiibaibiaiijiijii≥≤≤⋅=⋅∑∑≥≥,,0,!!∑∑∑∑∑-=-=-=-=-=⋅-≥⋅-≥⋅+⋅=≥⋅-=⋅-1010101010!)(!!1!!)(!)(jiiijiiijijijjjjiiiiabiabiiiipjbaiab0,10,0≥-≤≤⋅=∑≥ipapaniiii,,2,1,0,10,0tipapakitiiiL=-≤≤⋅=∑=13nk1n=0100·p111·p0n2,,2,1,0,10,110tipapakitiiiL=-≤≤+⋅=+∑=,11)1(1,110+=⋅=+⋅-=+-≡∑ttiiipppkpa10,)1(1)1(1},1{min,11100000000-≤≤⋅+⋅+=+⋅+⋅-=+-=-≠∃∑∑∑+==-=≤≤papapapappkpaiipaitiiiiiitiiiiiiiitii0,10,,,0,0≥-≤≤⋅=∃≤∀∑=ipapamunmmiuiii00,10,1,0≠≥-≤≤⋅=-∃∑=sisiiiaipapansssssiiispppppanp+-=≤+⋅=+=∑)1(1)1(0sspppn⋅-≤-)1(00,10,0≥-≤≤⋅=-∑=ipapapnisiiis0,10,0≥-≤≤+⋅=∑=ipappanissiii0,10,1,1110≥-≤≤⋅⋅=-=+=∑ipappanpaissiiis0,10,)1(,110≥-≤≤⋅+⋅=-≠∑=ipapapanpaisssiiis1,10,0≥-≤≤⋅=∑≥ipapanniiii0,1,0,00≥-≤≤⋅=⋅=∑∑≥≥ipbapbpaniiiiiiii141+jpjjba=}{max0iiibaij≠=≥iibai≠∃,jjba0,1,0,00≥-≤≤⋅=⋅∑∑==ipbapbpaiijiiijiii∑∑-=-=⋅-⋅-1010)()(jiiiijiiiipabpab}{min0iiibaij≠=≥jipbapbpaiijiiijiii≥-≤≤⋅=⋅∑∑≥≥,1,0,∑∑∑∑∑-=-=-=-=-=⋅-≥⋅-≥⋅-+⋅-=≥⋅-=⋅-1010101010)()1(1)1()()(jiiiijiiiijiijiijjjjjiiiipabpabppppppbapab15nixrxxxin,...,2,10,21=≥=+++LnrrC(n+r-1,r)5,0,1,3,2043214321≥≥≥≥=+++xxxxxxxx51344332211-==-=-=xyxyxyxy4,3,2,10,114321=≥=+++iyyyyyiC(4+11-1,11)C(14,11)364161rwyabcdrcrc0,c1,c2,c3,c41+3+4+3+1=122nmn=mnman{an}G(x)17(1-x)-mxn-m3852an8na0,a1,...,a8C(x)xkckk410094(+,-,×,÷)14nn18n-1n-1n-1n-14n-2anna2=1200991000,1,...,9,a0=1/2x2-10x-4005nnnann-1an-1n2(n-1)2(n-1)n2(n-1)2(n-1)2(n-1)19a0=26nnnann-1an-1n(n-1)(n-1)nnn⎩⎨⎧=+=-211anaanna0=1a2=40331212132132121211=-+-⇒⎩⎨⎧=+-=+-⇒⎩⎨⎧-=-=------------nnnnnnnnnnnnnnaaaaaaaaaanaanaax3-3x23x-10(x-1)3=0x=1nnCnBnAa1)(2⋅++=⎪⎩⎪⎨⎧===⇒⎪⎩⎪⎨⎧=++=++=⇒⎪⎩⎪⎨⎧===2121210144221421CBACBACBAAaaa1222++=nnan7PnD1,D2,...,Dnkn20ann1D1Dn-12D1Dn-1k-1D1Dn-1D1Dn-2n-2Dnk-2D1Dn-1n-1a1=0,a0=k⎩⎨⎧-==⇒⎩⎨⎧=--=+⇒⎩⎨⎧==110)1(010kBABAkkBAaka⎩⎨⎧-⋅-+-==1)1()1()1(1nkknkannn21222311nn24d0=1x2-x+1=012t⎩⎨⎧===--⇒⎩⎨⎧==+=⇒⎩⎨⎧==⎩⎨⎧=+=-------3,00323,02310231021101200111hhhhhhhhhhfhhfhfhttttttttttt