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max(N) 2 A8AB5<5, @02=> G8A;C ?0@, :>B>@K5 <>6=> A>AB028BL 87 N M;5<5=B>2 (N>1), B> 5ABL, @02=> G8A;C A>G5B0=89 87 N ?> 420: Smax(N) = N! /(2!(N-2)!)= (N-1) N / 2. (2) >MDD8F85=B s, >?@545;O5<K9 >B=>H5=85< s =S / Smax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isio.Drawing.11  8AC=>: 1  ;;NAB@0F8O 2=5H=53> 8 2=CB@5==53> 10;0=A8@>20=8O ?@>872>4AB25==>9 A8AB5<K 0:A8<0;L=>5 7=0G5=85 >F5=:8 A;>6=>AB8 7040G8 7028A8B >B N 8 @02=>: 0max =  Smax (1 -- 0) = (N-1)N/2. (6) A?>;L7CO <0:A8<0;L=CN >F5=:C (6), <>6=> ?>AB@>8BL >B=>A8B5;L=CN >F5=:C >1>1I5==>9 A;>6=>AB8 ?@8 7040==KE 7=0G5=8OE S 8 N: 0 >B= (S,N) = 0(Q0,S) / 0max(Q0,S). (7) =0G5=8O MB>9 >F5=:8 1C4CB ?@8=04;560BL 8=B5@20;C [0;1]. ;O @07;8G=KE F5;59 <>3CB 1KBL 8A?>;L7>20=K :0: 01A>;NB=0O, B0: 8 >B=>A8B5;L=0O >F5=:8 >@30=870F8>==>9 A;>6=>AB8. ;O ?>2KH5=8O 8=D>@<0B82=>AB8 01A>;NB=KE >F5=>: @5:><5=4C5BAO >3@0=8G8BL @07<5@=>ABL @0AA<0B@8205<KE A8AB5<, ?@8=O2, 8AE>4O 87 >A>15==>AB59 :>=:@5B=>3> :;0AA0 7040G, 2 :0G5AB25 25@E=59 3@0=8FK =5:>B>@CN E0@0:B5@=CN 4;O MB>3> :;0AA0 25;8G8=C Nmax .  MB>< A;CG05 ?>O2;O5BAO 2>7<>6=>ABL A@02=820BL ?> A;>6=>AB8 A8AB5<K @07;8G=>9 @07<5@=>AB8 2 ?@545;0E Nd"Nmax. $>@<C;0 4;O >F5=:8 CA;>2=>9 >@30=870F8>==>9 A;>6=>AB8 ?@8=8<05B 284: h(Nmax)>@3 =K S/ Smax(Nmax). (8)  MB>9 D>@<C;5 K  =5:>B>@K9 ?>;>68B5;L=K9 :>MDD8F85=B (Ke"1), 7040NI89 <0AHB01 H:0;K 87<5@5=8O CA;>2=>9 A;>6=>AB8, 2 G0AB=>AB8, ?@8=O2 =10, ?>;CG05< 4>AB0B>G=> C4>1=CN 45AOB810;;L=CN H:0;C, ?@8 K=100 >F5=:C ?>;CG05< 2 ?@>F5=B0E. ;O >F5=:8 CA;>2=>9 >?5@0F8>==>9 A;>6=>AB8 ?@8<5< 2K@065=85 h10; = K(1-Q0). (9) 1>1I5==K9 :>MDD8F85=B A;>6=>AB8 ?@8 CA;>288, GB> @07<5@=>ABL @0AA<0B@8205<KE A8AB5< >3@0=8G5=0 25;8G8=>9 Nmax, >?@545;5= :0: 2725H5==>5 ?@>872545=85 42CE ?>:070B5;59: h0 =  h(Nmax)10; 1- h(Nmax)>@3 =(K (1-Q0))1-(S/ Smax(Nmax)). 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( RpArialArial Arial"||`|]|| | ( @w|8`Xh=ؚ|| |w!+w @w (dv% TXUUAUUALP! % FEMF++@ @@4u?XLugDCQ DCQ D\(CDrMCQ DBQ DCugDC@$$==_888% % W8rDZ " " "$">"0Z 0% % $$AA( F4(EMF+ @$DԒBjB B( $$=='% % V0Bl"$"$ ) )"$% % $$AAFEMF+@@4u?H<DzQ0C 3'DzQ0C 3'DԒBDԒBDzQ0C@$$==_888% % V0@j"$ ) )"$"$ % % $$AA( FEMF+@@4u?H<1DԒB1D2B*D2B*D\(C 3'D\(C@$$==_888% % W0Wb%%B*B* ) % % $$AA( FEMF+@@4u?H<_D2B_DB*.D[VB*.DkC*DkC@$$==_888% % W0n[&B&++ * % % $$AA( FEMF+*@$BBD';C@0$*>ARIAL6@ >:C?0B5;8}=>#;>>>>Q>>>>y?>3?>B?>@yZ?>@r?>??   ( RpArialArial Arial"||`|]|| | ) @w|`Xh|ؚ|| |w!+w @w )dv% THUUAUUAH L`>:C?0B5;8  % FtEMF++@ *@$BBeC[B6@@4g=>??   % TT &8UUAUUA 6LP% FEMF++@ *@$BBeC[B@0$=ARIAL6@@42y>/>??   ( RpArialArial Arial"||`|]|| | * @w|`Xh=ؚ|| |w!+w @w *dv% TT.:UUAUUA8LP2% FEMF++@ @ ?CrMC?CCB C!8>Ci C,=Ci C!C:1C?CXC?CrMC@( $$=='%  ;UP             <> % $$AAF`TEMF+@@4u?@$$==_888% % UP             % % $$AA( FEMF+@@4u?H<=eCCUCCUC!B=eC!B=eCC@$$==_888% % V0n T^ ^ TT^ % % $$AA( FEMF+*@$BB=eC!B@0$*>ARIAL6@@4j=|>??   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