Tóm tắt lý thuyết 1. Định nghĩa Tổ hợp chỉnh hợp Hoán vị 2. nhị thức newton
– –!!–!"!"#$%&' ()*+,- ./- "0* ! (1*2'1345 ( ) ? n a b + = ( ) ? n a b − = !6'7- k k n n n C C − = 1 1 1 1 k k n n n C C k + + + = + 1 1 1 k k k n n n C C C + + + + = 8 5 9 : 7 5 9 : "; 5 9 '< => ?$, @ k nn k n C ,P ,A .AAB CD,0EF'9'G,H2'A'I,'J''''8,$- .'-CK'L2''58,$- ( ) M M B − = N OMN ( ) MM M − = !"#"$- 1 1 1 + + + =+ % P,9Q&Q$2'F',Q$2'R,''',$(S,$< 7; 1#-0@9'G,H2'=E6''G,, ( F' 5 &- ( ) 2 &CK,9Q ( ) n bxa + N ( ) n bxa 2 + (S,$<'J,AAT6' 1#EQ'J''8D,0E''=,$- (@9'G,H2''2, 0 n C 0E n n C UF' 0 2n C 0E n n C 2 2 VWX;'G, =EA<,,9Q ( ) ( ) nn x 2 x1 ;1 ++ (S,$<'JTO@$0@9'G,H2' L7$NF'TO@$'<7$ (@9'G,H2''2, 0 n C 0E n n C UF' 0 2 n C 0E n n C 2 2 VWX;'G, =E, ,'J,9Q ( ) ( ) nn x 2 x1 ;1 ++ (S,$'J a x ± = Y(@9'G,H2''2, 0 n C 0E n n C UF' 0 2n C 0E n n C 2 2 VWX;'G, '2, kkn ba − ,'J,9Q ( ) ( ) nn xa 2 xa ; ++ F' ( ) ( ) nn bxa 2 bxa ; ++ A,'- – –!!–!"!"#$%&' 'J ( ) ( ) T,0Z T,0Z +±= +±= 1 [(1@$EL'<F'99>,\6'9,0] 'CKE'7A'7 V.'Z'8- ^-.J*2'134Q,9Q ^-%W_E'7A'7`0@'G,,9Q ^Y-.J,AAT6' V1#F'9'G,EQ'J''8*2'- ,(a@9'G,H2'7 0 F' ( ) .F' 0 2 W9 X A ;0Z<F'8]$50* ,KE'7 (a@9'G,H2'7 0 F' ( ) .F' 0 2 W9 X ;0Z<=E6',$_>=>@ ,KE'7 '(a;''JI*2'134R''D,0EF'9'G,''[A,$ b=c''F'9'G,E YV.Id-e &E','<fF'9E,8\6'Q,0]E '<F'99Z8 ( )*+, V.'Z'8- ^ -.J ,9Q*2'134 ^ -%7_6'\,0@0Z'#6' ^ Y -66'\,0@ @?$8 V1#F'9'G,EQ'J''8 ,(a@9'G,H2''<'2, 0 0E UF' .0E 0 2 VWg$ 9XF'8&50* (ae X;9'<- −− − ,'J'#6'\=E '(a .J I * 2' 134 D, 0E '' [A ,$ b = '' F' 9'G,6'\ %*-./0 h#Ti- (a!6&Q$2''<'2, ,_'2&H2''<'2, ,$( (aSD',$j,'I=E`A@9Z''@?$8U2'=E]$8 '2VA'k`Y',@9Z''@?$8( Y(a @''#F'9'G,EQ'J''8''F' 9'G, C6l>'I,'/]'#@Y''8`E_=E''[A'' m0E''n`( &P,9Q*2'134U,^V NU,^TV (S,$<'J,AAT6' – –!!–!"!"#$%&' C[-! &F';'G,( 5- @;2'34Z?$- ( ) 1 0 n n k n k n k a b C a b − = + = !6o'G,T0E_ CD,0E8@Q6(!p<$_9,F';' Cm-! ;=Z79,9Q( 5- P,9Q*2'34Z?$( )F k a =E;?$'G,9,9Q Ti65;$'G,b_ { } k a (!p<Q$_9, ( ) max k a .Id-QTi65;$'G, { } k a 0Z k N ∈ ,ql- • Tir- 1 k k a a + • si;$- 1 k k a a + − – –!!–!"!"#$%&' 8 5 9 : 7 5 9 : "; 5 9 '< => ?$, @ k nn k n C ,P ,A 123++-!@E3''Z',$- 4 25)F]$;'<+,=E- ∈ ≥ Nkn kn , 4 25-CK'''L2',$Q9IJ- ( ) n k n P. ! ! A kn n − = O !n A ( ) !! ! C k n knk n − = 4 25S,$9IJ'59 :759 :;59 b '0]59 :759 :;59 '58(8 ;0E 'J;6'`]$;`Z' 4 25&P@=$# 6)0Z;59 ,'<Q835Ftl 4 7 8''59 ,$ ,V 13 5 nn CC = V 1 14 2 1414 2 ++ =+ nnn CCC 8 ,V]$;- Znn ∈ ≥ ,0 – –!!–!"!"#$%&' ( ) ( ) ( )( ) 3012-n !1 ! 5 !3-n3! n! 5 13 =−⇔ − =⇔= n n n CC nn ( ) = = ⇔=−−⇔ =[ u 0283 #_;'G,59 =EOu V]$;- ∈ ≤ ⇔ ∈ +≥ ++ v 214 UhV !,'<- 1 14 2 1414 2 ++ =+ ( ) ( ) ( ) ( ) ( ) MM M MM M M[M [M −+ = −+ +⇔ 131 14 2 122 14 ( )( ) ( )( ) ( )( ) −+ = ++ + − ⇔ 131 2 21 1 14Y ( ) ( ) ( ) ( ) ( ) ( ) − + = − − + + + ⇔ 142214132 = = ⇔=+−⇔ 8 03212 [ 8UhV #_;'G,59 =E- = = 8 4 4 7 8''59 ,V B 43 2 1 =+ + V ( ) 2 32 1 4 BB =−− + 8 ,V]$;- ∈ ≥ + 2 !,'<- B 43 2 1 =+ + ( ) ( ) ( ) M M MM M Y 2 42 12 − =+ − + ⇔ ( ) ( ) 184 − = + + ⇔ Y ( ) ( ) = = ⇔=−⇔ # = m 3 2 015 V]$;- ∈ ≥ + 2 8@ ( ) ( ) ( ) ( ) ( ) 2 3 12 2 4 212 − =− − − − + ⇔ M M M M MM M ( ) ( ) [[ Y = + ⇔ 2 0310 2 =−+⇔ Y – –!!–!"!"#$%&' =−+ = ⇔ 03108 0 2 59 E_0L; 2 ≥ ∀ #_59 b'0L; 4 7'"b_ $_>58bH2' 0 4 5 3 1 4 1 =−− −− B 8 ]$;- ≥ ∈ 5 59 b'550Z- ( ) ( ) ( ) ( ) ( ) ( ) 0 4 2 4 5 43 1 54 1 = − − − − − − − − M M ( MM M MM M ( ) 0 4 1 4 5 46 1 = − − − − −⇔ [ ( ) ( ) ( ) 065414 = − − − − ⇔ ( 0229 =−−⇔ ( ) −= = ⇔ = 2 11 ⇔ O=E9*' 4 78! A ∈ v ^ @- 355 1 11 1 1 = − ++ + + 8 !,'<- ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 + −+ = −+ + −+ + = + + + MM M MM M E- ( ) ( ) ( ) ( ) ( ) 2 21 1 1 1 1 1 1 +− = +−− + −+ + = − + + MM M MM M !p8@ = +− = + −+ 3 52 1 1 1 ⇔ =+− = 0683 2 ⇔ + ∈ = = 3 6 #_ = = 6 3 =E''9*' 4 78''59 ,$- ,V 10 3 2 2 1 ≥ + – –!!–!"!"#$%&' V ( ) 114 1 1 3 +<+ − ++ B 8 ,V]$;- ∈ ≥ + 2 ( ) ( ) ( ) MM M ( MM M 2210 3 12 1 10 3 2 2 1 − ≥ − + ⇔≥ + ( ) 501013 2 1 10 3 2 1 2 ≤≤⇔≤−−⇔ − ≥ + Y Y M C ∈ ≥ + 2 >OwAYA[Amx #_'';'G,759 =E-OwAYA[Amx V]$;- ∈ ≥ + 2 ( ) ( ) ( ) ( ) ( ) ( ) 114 21 1 2 114 1 1 3 +< − + + − + ⇔+<+ − ++ MM M M M B ( ) 14 2 1 <+− ⇔ 40282 2 <<⇔<−− u C + ∈ 0E 2 ≥ >OwAYx #_'';'G,759 =E-OwAYx 4 7%8''759 ,$- ,V ( ) [ M B 4 143 4 < + + UV V 24 23 4 ≤ − − + Y [ B B UV 8 ,V]$;- ∈ 1 UV ⇔ ( ) ( ) M MM M 2 4 1434 +< + ⇔ ( ) ( ) 143434 < + + ⇔ 095284 2 <−+ ⇔ y(mz z(m C ∈ > { } A{A = #_'';'G,759 =E- { } YAA = V]$;- ∈ ≥ + 4 UV ⇔ ( ) ( ) ( ) ( ) 24 23 442 1 4 ≤ − − − + − MM M M M M M – –!!–!"!"#$%&' ⇔ ( )( ) 24 23 1 3 4 4 ≤ − − + M M ⇔ 0564 2 ≤+− ⇔ 51 ≤ ≤ C ∈ ≥ + 4 > { } mA4 = #_'';'G,759 =E- { } mA4 = 4 7&8759 - 4 5 3 1 4 1 −− −− 0≤ T B UV 8 ]$;- ∈ ≥ + 5 UV ⇔ ( ) ( ) ( ) ( ) ( ) ( ) 0 4 2 4 5 43 1 54 1 ≤ − − − − − − − − M M MM M MM M ⇔ ( ) 0 4 1 4 5 43 1 4 1 ≤ − − − − − − MM ⇔ 0229 2 ≤−− ⇔ 11 ≤ ≤ #_#;'G,759 =E- { } {AyA|AuAnAmA = 4 798759 ( ) −≤++++ 742 321 ((( ( ) 1 ≥ UV 8 !,'<- ( ) ++++=+ ((( 2210 1 UV %7_E'7,0@'G,UV,'- ( ) 1211 1 21 − − ++++=+ ((( UV .JTO,'- ++++= − (((( 3211 322 P<-UV ⇔ ( ) −≤ − 72 } ( ⇔ ( ) −≤ − 72 1 ⇔ 72 1 ≤+ − UV "E += V~U '< =VU~• ≥∀>+= − 0122 1 ~UV=EE 1 ≥ ∀ %'<-~UYVOu ;'G,759 UV=E- 3 ≤ #_#;'G,759 b'=E- { } YAA1 = .AAB CD,0EF'9'G,H2'A'I,'J''''8,$- .'-CK'L2''58,$- ( ) M M B − = N OMN ( ) MM M − = – –!!–!"!"#$%&' 4 7.29R 1 1 + ++ + = + ∈ A !,'<- ( ) ( ) ( ) ( ) + + + = + = −+ + + = + MM M MM M 11 11 1 ' 4 7.AA ∈ 10E ≤ N ≤ (.29R- − − = 8 !,'<- ( ) ( ) ( ) ( ) ( ) ( ) MM M ( MM M MM M ( MM M −− = −− − − = − − O ( U'V 4 7%. + ∈ (.29R- 1 22 1 22 2 1 + + − −+ UV 8 .',-!3'L2'- 1 1 1 + + + =+ ,'<- 12 1 22 + − =+ P<-UV ⇔ ( ) ( ) ( ) ( ) ( ) ( ) MM M MM M ( 112 2212 1 12 2 1 122212 ++ + + = + + == +++ O ( ) ( ) ( ) 1 22 2 1 112 22 + + = ++ + MM M ' .'-P,9Q0@9(!,'<- BO ( ) ( ) ( ) ( ) MM M MM M 11 22 1 22 +− +=+ − O ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) MM MM MM MM 11 2221222 2 1 11 1212 2 ++ ++++ = ++ +++ O ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 22 2 1 11 22 2 1 11 22122 2 1 + + = ++ + = ++ + + MM M MM M U'V 4 7&. 2 ≥ 0E + ∈ (.29R- += + 22 1 UV 8 UV ⇔ 122 1 += + !,'<- ( ) ( ) ( ) ( ) MM MM M M MM M 12 21 122 12 − + − = − + − =+ O ( ) ( ) ( ) ( ) 2 1 12 1 12 1 + = − + = + + M M MM M U'V 4 7'.29R – –!!–!"!"#$%&' ( ) 2 1 32 112 3 1 2 1 + =++++++ −− (((((( 8 !,'<- = 1 ( ) ( ) ( ) ( ) 1 1 22 2 1 22 22 1 2 −= − − = − − = M M ( MM M ( M M MM M (( ( ) ( ) 2 22 33 33 2 3 −= − − = MM M MM M (( €€€€€€€€€(( 1 1 +−= − ( €€€€€€€€€( 1 1 = − ( .&H2'9>0@30@,'- 112 3 1 2 1 32 =− ++++++ (((((((((( O ( ) ( ) ( ) 12121 + + + + − + + − + − + (((((( O ( ) 2 1 321 + =++++ ((( U!3'L2'6'7'&V 4 7(.29R 2 1 1 2 1 2 3 4 2 2 3 1 1 2 0 =++++++ + + + +++++ (((((( 8 !,'<- ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) MMM MMM MM M - MM M −++ ++ = ++ ++ − = + ++ 2 11 11 2 1 2 O ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +−− + − ++− + = −++ ++ MM MM MM MM MM MM 21 1 1 1 2 1 2 11 O 12 1 1212 2 1 + −− + − + − ( UhV %=,_TO{AAAYA€A0EUhV,'-