ࡱ> uwt5@bjbj22,XXA-8,"3F.B"D"D"D"?"o*[2$h4R62]22 l"@ B" !! W^z!!20"3!>7>7!>7!$ 22 (>@=K5 D>@<C;KCABL 8<55BAO 65AB2>, A>AB>OI55 87 n M;52. 1>7=0G8  EMBED Equation.3 . 5@5AB0=>2:>9 87 n M;52 =07K205BAO 7040==K9 ?>@O4>: 2> 65AB25  EMBED Equation.3 .@82>::1)@0A?@545;5=85 n @07;8G=KE 4>;6=>AB59 A@548 n G5;>25:;2)@0A?>;>65=85 n @07;8G=KE ?@542 2 >4=>;L:> @07;8G=KE ?5@5AB0=>2>: 6=> >1@07>20BL 2> 65AB25 EMBED Equation.3 ? '8A;> ?5@5AB0=>2>: >1>7=0G05BAO Pn (G8B05BAO   87 n ).'B>1K 2K25AB8 D>@2>:, ?@54AB028=C20==KE G8A;02:8 1C451@07>2K20BL, @0A?>;030O M;5<5=BK Un 2 MB8E OG59:0E.  ?5@2CN OG59:C 6=> 70=5AB8 ;N1>9 87 n M;52 (8=0G5: ?5@2CN OG59:C 6=> 70?>;=8BL n @07;8G=KA>10;=82 ?5@2CN OG59:C, 6=> =09B8 n 1 20@80=B>2 70?>;=5=8O 2B>@>9 OG59:8. "0:81@07>2 70?>;=5=8O 42CE ?5@2KE OG55:. @8 70?>;=5=88 ?5@2KE 42CE OG55: 6=> =09B8 n 2 20@80=B0 70?>;=5=8O B@5BL59 OG59:8, >B:C40 ?>;CG05BAO, GB> B@8 OG59:8 6=> 70?>;=8BL n(n-1)(n-2) A?>A>104>;60O MB>B ?@>F5AA, ?>;CG8 G8A;> A?>A>1>2 70?>;=5=8O n OG55: @02=>  EMBED Equation.2 . BAN40 Pn = n(n  1)(n  2)...(3(2(1'8A;> n(n  1)(n  2)...(3(2(1, B> 5ABL ?@>872545=85 2A5E =0BC@0;L=KE G8A5; >B 1 4> n, =07K205BAO "n-D0:B>@80;" 8 >1>7=0G05BAO n! BAN40 Pn =n!> >?@545;5=8N AG8B05BAO: 1!=1; 0!=1.@8;L:> ACI5AB2C5B 20@80=B>2 70;6=>AB59 5-N :0=4840B0<8?  EMBED Equation.2  . 07<5I5=8O<8 87 n M;52 ?> k M;52 1C45@O4>G5==K5 ?>465AB20, A>AB>OI85 87 k M;52 65AB20  EMBED Equation.3  (65AB20, A>AB>OI53> 87 n M;52). '8A;> @07<5I5=89 87 n M;52 ?> k M;52 >1>7=0G05BAO  EMBED Equation.2  EMBED Equation.2  (G8B05BAO " 87 n ?> k").4=> @07<5I5=85 87 n M;52 ?> k M;52 65B >B;8G0BLAO >B 4@C3>3> :0: =01>@>< M;52, B0: 8 ?>@O4:>;>65=8O. @84OI8E : =5>1E>48AB8 ?>4AG5B0 G8A;0 @07;L:8A>10<8 6=> 2K1@0BL 87 15 G5;>25: 5 :0=4840B>2 8 =07=0G8BL 8E =0 5 @07;8G=KE 4>;6=>AB59?2) !:>;L:8A>10<8 6=> 87 20 :=83 >B>1@0BL 12 8 @0AAB028BL 8E 2 @O4 =0 ?>;:5? 7040G0E > @07;0305BAO k ;53:> ?>;CG8BL EMBED Equation ;O ?>4AG5B0  EMBED Equation.2  8A?>;L7C5B 65 4, GB> 8A?>;L7>20;AO 4;O ?>4AG5B0 Pn, B>;L:> 745AL 2>7L<5< ;8HL k OG55:. 5@2CN OG59:C 6=> 70?>;=8BL n A?>A>10@CN, ?@8 70?>;=5==>9 ?5@2>9, 6=> 70?>;=8BL n 1 A?>A>10<8. "0:81@07>2 70?>;=5=8O ?5@2KE 42CE OG55:. >6=> ?@>4>;60BL MB>B ?@>F5AA 4> 70?>;=5=8O ?>A;54=59 k 9 OG59:8. -BC OG59:C ?@8 70?>;=5==KE ?5@2KE k 1 OG59:0E 6=> 70?>;=8BL n (k 1) (8;8 n k+1) A?>A>10<8. "0:81@07>;=ONBAO G8A;>A>1>2, @02=K;CG05<: EMBED Equation @8;L:> ACI5AB2C5B @07;8G=KE 20@80=B>2 2K1>@0 4-E :0=4840BC@ 87 9-B8 A?5F80;8AB>2 4;O ?>574:8 2 4 @07;8G=KE AB@0=K?  EMBED Equation.2 !>G5B0=8O<8 87 n M;52 ?> k M;52 =07K20NBAO ?>465AB20, A>AB>OI85 87 k M;52 65AB20  EMBED Equation.3  (65AB20, A>AB>OI53> 87 n M;52).4=> A>G5B0=85 >B 4@C3>3> >B;8G05BAO B>;L:> A>AB02>< 2K1@0==KE M;52 (=> =5 ?>@O4:>;>65=8O, :0: C @07 A>G5B0=89 87 n M;52 ?> k M;52 >1>7=0G05BAO  EMBED Equation.2  (G8B05BAO "C 87 n ?> k").@84OI8E : ?>4AG5BC G8A;0 A>G5B0=89: 1) !:>;L:> ACI5AB2C5B 20@80=B>2 2K1>@0 6-B8 G5;>25: 87 15 :0=4840B>2 4;O =07=0G5=8O =0 @01>BC 2 >48=0:>2KE 4>;6=>ABOE?2) !:>;L:8A>10<8 6=> 87 20 :=83 >B>1@0BL 12 :=83?K2545@4AG5B0 G8A;0 A>G5B0=89. CABL 8<55BAO 65AB2>  EMBED Equation.3  8 =C6=> >1@07>20BL C?>@O4>G5==>5 ?>465AB2> 65AB20  EMBED Equation.3 , A>45@60I55 k M;52 (B> 5ABL >1@07>20BL @07<5I5=85). 5;05 B0:: 1) 2K45;8 k M;52 87 n M;52 65AB20  EMBED Equation.3  -B>, A>3;0A=> A:070==><C 2KH5, 6=> A45;0BL  EMBED Equation.2  A?>A>10@O4>G8< 2K45;5==K5 k M;52, GB> 6=> A45;0BL EMBED Equation  A?>A>10 6=> ?>;CG8BL EMBED Equation  20@80=B>2 (C?>@O4>G5==KE ?>465AB2), >B:C40 A;54C5B: EMBED Equation  , B> 5ABL  EMBED Equation.2  (1)@825: 87 15 6=> 2K1@0BL G8A;>A>1>2, @02=K6=> ?>=OBL, GB> >ACI5AB28BL 2K1>@ ?>465AB20 87 B M;52 65AB20, =0AG8BK20NI53> ? M;52, 6=>, 2K1@02 ? B M;52, :>B>@K5 =5 2>94CB 2 8=B5@5ACNI55 =0A ?>465AB2>. BAN40 A;54C5B A2>9AB2> G8A;0 A>G5B0=89  EMBED Equation.3 -BC D>@<C;C 6=> 4>:070BL, 8A?>;L7CO D>@4AG5B G8A;0 ?>465AB2 :>=5G=>3> 65AB20 =07K20NBAO :>@=K<8. 0AAB@8B>@K5 :>@=K5 7040G8.1.7 A54>2 >@30=870F8O 4>;6=0 2K1@0BL B@8 4;O @072. !:>;L:8A>10<8 6=> @0728O OA=>, GB> :064K9 702>4 65B ;81> ?>;CG8BL >48= 70:07, ;81> =5 ?>;CG8BL =8 >4=>3>, 8 GB> 2K1@02 B@8 702>40, 6=> ?>-@07=><C @07 AG8B0BL G8A;> @07285 @07;8G8O B@5E 70:07>2, A>E@0=82 2A5 >AB0;L=K5 CA;>28O, ?>;CG8A>1 @072 >?@545;O5BAO B>;L:> 2K1>@>9:8 702>4>2, B0: :0: 2A5 MB8 702>4K ?>;CG0B >48=0:>2K5 70:07K, 8 G8A;> 20@80=B>2 >?@545;O5BAO :0: G8A;> A>G5B0=89.  EMBED Equation.2  EMBED Equation.2 3.4>2. !:>;L:8A>10@30=870F8O 65B @07872>4AB25==KE 70:070? (0:07 =5;L7O 4@>18BL, B> 5ABL @0A?@545;OBL 53> =0 =5A:>;L:8E 702>40E). >B;8G85 >B CA;>28O ?5@2>9 7040G8, 745AL >@30=870F8O 65B >B40BL 2A5 B@8 70:070 ?5@2>4C 8;8, =0?@8B40BL 420 70:070 2B>@>4C, 0 >48= - A54L<C.040G0 @5H05BAO B0:. 5@2K9 70:07 65B 1KBL ?><5I5= A5<LN @07;8G=KA>1045, =0 2B>@><5AB82 ?5@2K9 70:07, 8<55< A52 ?>@>3> (8=0G5, :064K9 A?>A>1 ?>3> 70:070 65B A>?@>2>640BLAO A5A>10@>3>). "0:81@07>A>1>2 @072. 07<5AB82 8E :0:8 >1@07><, 652 ?> (8=0G5, :064K9 A?>A>1 @072 65B A>?@>2>640BLAO A5<LN @07;8G=KA>10 70:070). !;54>20B5;L=>, ACI5AB2CNB 49(7=73 A?>A>1>2 @072. (A;8 1K 70:07>2 1K;> n, B> ?>;CG8;>AL 1K 7n A?>A>1>2 @07<5I5=8O).4.0: @5H0BL 7040GC 3, 5A;8 2 55 B5:AB5 2 A;>2 "@07;8G=KE ?@>872>4AB25==KE 70:070" ?>AB028BL ">48=0:>2KE ?@>872>4AB25==KE 70:070"? -B> B@C4=0O 7040G0. 865 ?@82>48BAO 0=0;>38G=0O 7040G0 040G0 V A @5H5=85102828N 7040G8 1 >4=C D@07C: >@30=870F8O B0:65 4>;6=0 @0A?@545;8BL B@8 @07;8G=KE 70:070 =0 873>B>2;5=85 45@52O==KE ?5@5:@KB89 A@548 4-E ;5A>?8;>:. !:>;L:8A>10<8 3CB 1KBL @0A?@545;5=K 2A5 70:07K?064K9 87  EMBED Equation.2  A?>A>1>2 @0A?@545;5=8O 70:07>2 =0 702>40E 65B A>?@>2>640BLAO  EMBED Equation.2  A?>A>10<8 @072 =0 ;5A>?8;:0E. 1I55 G8A;> 2>76=KE A?>A>1>2 @072 1C45B @02=>  EMBED Equation.2 6. 8M;B5@A:0O D8@406C 5 1>;LH8E :20@B8@ 8 4 3010@8B=KE :20@B8@K. 0=: =0<5@5205BAO :C?8BL 4:20@B8@K, ?@8GQ;6=> 1KBL 1>;55 42CE 3010@8B=KE. !:>;L:> 20@80=B>2 2K1>@0 8<55B 10=:?0=: 65B :C?8BL 4 1>;LH85 :20@B8@K. # =53> 5ABL 2>76=>ABL 2K1@0BL 4 87 5-B8 ?@54;0305 20@80=B>2 745AL @02=>  EMBED Equation.3 . A;8 10=: @5H8B :C?8BL B@8 1>;LH85 :20@B8@K 8 >4=C 3010@8B=CN, B> G8A;> 20@80=B>2 2K1>@0 C =53> 1C45B @02=>  EMBED Equation.3 . A;8 1C45B ?@8=OB> @5H5=85 :C?8BL 425 3010@8B=KE :20@B8@K 8 425 1>;LH8E :20@B8@K, B> G8A;> 20@80=B>2 1C45B @02=K<  EMBED Equation.3 . "0:81@07>2 2K1>@0. 040G8 A @5H5=8O;L:8<8 @07;8G=KA>10<8 6=> @0AAB028BL =0 ?>;:5 A>1@0=85 A>G8=5=89, A>AB>OI55 87 10-B8 B>2, ?@8 CA;>288, GB> ?5@2K9 8 ?OBK9 B>;6=K AB>OBL @O4>:><18=0B 818;59 9 3@C7>?>4JQAB8 8 10 1>;LH53@C7=KE 02B>18;59. 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EMBED Equation.3 ( EMBED Equation.3 . 040G0 III.  EMBED Equation.3 +2( EMBED Equation.3 . 040G0 IV. (? 1)! 040G0 V.  EMBED Equation.3 . 040G0 VI.  EMBED Equation.3 . 040G0 VII.  EMBED Equation.3  040G0 VIII. 57. 5H5=8O.040G0 I.A53> ACI5AB2C5B 10! @07;8G=KE ?5@5AB0=>2>: 10-B8 :=83. 'B>1K ?>4AG8B0BL, A:>;L:> 6=> =09B8 ?5@5AB0=>2>:, 2 :>B>@KE ?5@2K9 8 ?OBK9 B>OB @O4>;>68 : ?5@2>=8 :0: 1K >1@07CNB >B45;L=CN :=83C. "0:81@07>;CG8;>AL 9 :=83, :>B>@K5 3CB 1KBL @0AAB02;5=K 9! A?>A>10 CG5ABL, GB> ?5@2K9 8 ?OBK9 B><0 3CB 1KBL A:;55=K 2 4@C3>@O4:5, 8 6=> ?>;CG8BL 5IQ 9! @07;8G=KE ?5@5AB0=>2>: 10-B8 :=83, 2 :>B>@KE ?5@2K9 8 ?OBK9 B>OB @O4> >B25B 7040G8 A>AB02;O5B G8A;>, @02=>5 10! 2(9!=8(9! 040G0 II.48= 2K1>@ B@>9:8 02B>18;59 9 3@C7>?>4JQAB8 >B 4@C3>3> 65B >B;8G0BLAO =5 B>;L:> A>AB02>< 2K1@0==KE  8 8E @0A?@545;5=85 B>@3>2K@30=870F8O76=>, GB> MB8 B>@3>2K5 >@30=870F88 @0A?>;>65=K =0 @07;8G=KE @0AAB>O=8OE >B 02B>:> C =8E @07=K5 CA;>28O >?;0BK B@C40 8 B. ?. 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