Toán 11 xác suất

Gv: Lê Tuấn Cảnh

 !"#
Phương pháp giải toán trong phép đếm
B1. Phân tích và quản lí giả thiết
Chẳng hạn có m cách chọn đối tượng a và n cách chọn đối tượng b, cần phân biệt:
TH1: Bất kỳ cách chọn nào trong m cách chọn đối tượng a đều không trùng ( không phụ
thuộc) với bất kỳ cách chọn nào đó trong n cách chọn đối tượng b => sử dụng quy tắc cộng m
+ n cách chọn thoả ycbt
TH2: Bất kỳ cách chọn nào trong m cách chọn đối tượng a đều tương ứng ( phụ thuộc) vào bất
kỳ cách chọn nào đó trong n cách chọn đối tượng b => sử dụng quy tắc nhân m.n cách chọn
thoả ycbt
TH3: Khi bài toán phép đếm phức tập, ta kết hợp cả hai quy tắc và ghi nhớ: Phải thực hiện
quy tắc nhân trước và quy tắc cộng sau =>m
1
.n
1
+ m
2
.n
2
cách chọn theo ycbt.
B2. Thực hiên các phép tính cụ thể => ycbt
!$%
&'()%*+,-./%*0&(%01,-2$(%01,-03(&45+067
89:;%,-./%*0&(%01,-45+060&<=>$
ab
?@A%
{ }
? BCCDCECFa b ∈

@$
Ba ≠
;(0&2

GD0H00;@$0;'0&2

GI0H00;H00;(%4-%
.J(@$'0&K+(LMN.+O0?*.(0&9G2

2

GDIGB,-.)PQ0'.
!$%
).RM5
{ }
CCDCICSB =
&.T>RM4J0.U!9
(V !()%*+,-W2D01,-XH0(+7
'V !()%*+,-06W2D01,-XH0(+7
0V !()%*+,->YW2D01,-XH0(+7
89
(V :;%,-W2D01,-XH0(+>$
abcd
CX%4&0;0H04-%.J
? ? ?a b c d B∈
;
{ }
CCDCICSa B∈ =
GZ0&2


GI0H00;
;
{ } { }
CCDCICS [b B a∈ =
GZ0&2

GD0H00;
;
{ } { }
CCDCICS [ ?c B a b∈ =
GZ0&2
\
G\0H00;
;
{ } { }
CCDCICS [ ? ?d B a b c∈ =
GZ0&2
D
G0H00;
T]9!-0H00;0H04-%.J^(C'C0C<V_.`*0&K+(LMN.+O0?*'$%.)H
%P%.a)K+Q.b0cRQ0&G2

2

2
\
2
D
GID\GB,-.)PQ0'.
'V `*!0H0,-06.$>RM4J0MP%.R0d>$)e0Df.0H0.`gJM,(+9

8=,-0h.i2>$
abc
j%4&0;
{ }
? ? CDCICSa b c ∈
?0;(0&D0H0?0;'
0&\0H0@$00&0H00;./.`*0&

GD\GD,-06.R0d
'k
8=,-0h.i2>$
Dabc
j%4&0;
{ }
? ? CCICSa b c ∈
?0;(0&D0H0?0;'
0&\0H0@$00&0H00;./.`*0&

GD\GD,-06.R0d
'kD
H00;0H04-%.J
( )
C Dabc abc
0&K+(L4O0>RM(+?*.(Xl.JM@$
(0&9G

m

GDmDGDF,-.)PQ0'.
0V (4n0&9GB,-W2'-01,-XH0(+.`*!@$oGDF,-W2D01,-06

XH0(+.`*!?*.(0&9pGqoGBqDFGS,->Y.)PQ0'.^M<N0H0
%P%0c+'../0)0c+0?.(0r0&Xl.K+PS,-V
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(1

Gv: Lê Tuấn Cảnh
!$%\
).RM5
{ }
BCCC\B =
&.T>RM4J0.U!9
(V !()%*+,-W2D01,-XH0(+7^,9F,-.)PQ0'.V
'V !()%*+,-06W2D01,-XH0(+7^,9B,-.)PQ0'.V
0V !()%*+,->YW2D01,-XH0(+7^,9F,-.)PQ0'.V
89M<N0H0%P%'$%?>+]9;,-0h.i2
abcd
.i
Ba
≠
!$%D
O.Xl.,b.0&It2X)H`%*'%L.?2u%t2X)H45+0&@v4H,-BCCC\CDCICECSCFCw
O.<nQI01,-0)2O.0H02_Xf.&'()%*+MH2_Xf.XH0(+7
89:;%
abcde
>$2O.MH2_Xl..+x]0h.i2T]0H04-%.J(?'?0?<?a0&,-
0H00;(+_.U@s.`s.y./2$&4y@$0H00H00;4-%.J$Q0&
K+(LMN.+O0
;
{ }
BCCC\CDCICECSCFCwa ∈
0&B0H00;?..@RQ'?0?<?a45+0&B0H0

0;#@RQ,-0h.i2>$9GB
I
GBBBBBMH2_Xf.
!$%I
&'()%*+,-W2'(01,-.`)4&0z0&4t01,-I7
89:;%,-0h.i20&<=
abc
@$
{ }
? ? BCCC\CDCICECSCFCwa b c ∈
T,-.)PQ0'.0&'(
XP{|PQ`(9
H0,-0&<=
I C^ I? IVbc b c≠ ≠
?X%4&.(0&w0H00;'@$w0H00;'
RQ0&wwGF,-<=
Ibc
H0,-0&<=
{ }
I C^ BCI ? IVa c a c≠ ≠
?X%4&.(0&F0H00;(@$w0H00;0
RQ0&FwGS,-<=
(Ic
\H0,-0&<=
{ }
IC^ BCI ? IVab a b≠ ≠
?X%4&.(0&F0H00;(@$w0H00;'
RQ0&FwGS,-<=
('I
&2>=%.(0&9FmSmSGI,-.)PQ0'.

!$%E
&'()%*+,-./%*W2I01,-2$0H001,-45+>AD@$4}%2O.XH0(+7
89:;%<=,-0h.i2>$
abcde
?
{ }
? ? ? ? I?E?S?F?wa b c d e B∈ =
;
a B∈
0&2

GI0H00;
;
{ }
[b B a∈
0&2

GD0H00;
;
{ }
[ ?c B a b∈
0&2
\
G\0H00;
;
{ }
[ ? ?d B a b c∈
0&2
D
G0H00;

;
{ }
[ ? ? ?e B a b c d∈
0&2
I
G0H00;
RQ0&G2

2

2
\
2
D
2
I
GID\GB,-.)PQ0'.
!$%S
&'()%*+,-W2D01,-XH0(+4J0.()*.U0H001,-\?I?S?F7
89:;%<=,-0h.i2>$
abcd
?
{ }
? ? ? \?I?S?Fa b c d B∈ =
;(0&D0H0?0;'0&
\0H0?0;00&0H0@$0;<0&0H0RQ0&D\GD,-.)PQ0'.
!$%F
A%B01,-.UB4lw?0&.T>RM4J0'()%*+,-060&D0~,-2$0H001,-4&45+
XH0(+7
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(2

Gv: Lê Tuấn Cảnh
89:;%D,-0h.i20&<=
  \ D
a a a a
?

C B
i j
a a a≠ ≠
C(
D
06*
{ }
D
BCCDCECFa E∈ =
j%4&
.(0%('$%.)H.$I.`gJM0H0,-0&<=9
  \   \   \   \   \
BC C DC EC Fa a a a a a a a a a a a a a a
9
  \
Ba a a
?0;(

0&w0H0?0;(

0&F0H0?0;(
\
0&S0H0RQ0&wFSGIBD

0H00;,-0&<=
  \
Ba a a

./@RQ0)0H0?\?D?I@$0&0d2O.Xl.K+P9DDF0H00;
RQ.(0&9IBDmDDDFGwE0H00;,-.)PQ0'.
!$%w
)F01,-BCCC\CDCICECSUF01,-.`*0&.T>RM4J0'()%*+,-?2u%,-W2D01,-
4}%2O.XH0(+@$X}0%(l.0)B7
89:;%D,-0h.i20&<=
  \ D
a a a a
?

C B
i j
a a a≠ ≠
`)4&
{ }
  \ D
? ? ? BCCC\CDCICECSa a a a B∈ =
@$<)'-,-X}0%(l.0)B*
D
Ba ≠
;
D
a B∈
0&S0H00;?0;
{ }
 D

[a B a∈
0&E0H00;?0;
{ }
 D 
[ ?a B a a∈
0&E0H0
0;?0;
{ }
\ D  
[ ? ?a B a a a∈
0&I0H00;
RQ0&9SEEIGEB0H00;,-.)PQ0'.
!$%B
UI01,-BCC\CICS0&.T>RM4J0'()%*+,-?2u%,-W2D01,-XH0(+@$X}0%(
l.0)I7
89:;%D,-0h.i20&<=
  \ D
a a a a
?

C B
i j
a a a≠ ≠
`)4&
{ }
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{ }
D

BCIa ≠

;
{ }
D
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0&\0H00;?0;
{ } { }
 D
C\CI [a a∈
0&\0H00;?0;
{ }
 D 
[ ?a B a a∈
0&\0H00;?0;
{ }
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[ ? ?a B a a a∈
0&0H00;
RQ0&9\\\GID0H00;,-.)PQ0'.
!$%
&'()%*+,-06W2E,-XH0(+4}%2O..`)4&01,-4h+.%*>$01,->Y7
89:;%,-0&E01,-0h.i20&<=9
  \ D I E
a a a a a a
?
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C B
i j
a a a≠ ≠

?.`)4&
{ }
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? ? ? ? ? BCCC\CDCICECSCFCwa a a a a a B∈ =
8)01,-4h+.%*>$,->Y*
{ }

?\?I?S?wa ∈
@$@i>H,-06*
{ }
E
BCCDCECFa ∈
j%4&9;(

0&I0H00;?
0;(
E
0&I0H00;?0;(

0&F0H00;?0;(
\
0&S0H00;?0;(
D
0&E0H0
0;@$0;(
I
0&I0H00;
RQ.(0&9IIFSEIGDBBB,-0;.)PQ0'.
!$%
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(3

Gv: Lê Tuấn Cảnh
)I01,-BCCC\CDUI01,-4&0&.T>RM4J0'()%*+,-060&I01,-,()0)
.`)2u%01,-4&?2u%01,-.`*0&2e.4t2O.>h7
89:;%,-0h.i20&<=>$
abcde
?
{ }
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^
Ba
≠
@$a>$,-06
*
{ }
BCCDe ∈
j%4&.(|f.\.`gJM03(a
-0&<=
Babcd
;
{ }
? ? ? CC\CDa b c d B∈ =
.i.(0&9D\GD,-06
<=
Babcd
-0&<=
abcde
?
{ }
CDe ∈

0&0H00;?0;
{ } { }
CC\CD [a B e∈ =
0&\0H0
0;?0;
{ } { }
BCCC\CD [ Cb B e a∈ =
0&\0H00;?0;
{ } { }
BCCC\CD [ C Cc B e a b∈ =
0&0H00;@$0;
{ } { }
BCCC\CD [ C C Cd B e a b c∈ =
0&0H00;RQ9\\G
\E
RQ0&9Dm\EGEB,-.)PQ0'.
!$%\
O..`g.%T+;00&IB;0,%4=.<(%L+0H+)(!H0W?.`)4&0&'-0eM(a2
,%4}%#$.`g0h0;2O.&2\;0,%.`)IB;0,%.`*</=%O%0H+)(
!H0W,()0).`)&2X}0&0eM(a2,%4}%$)•%0&'()%*+0H00;7
89O.&2\;0,%,()0)X}0&0eMa2;0,%,%4}%$)?*.(0&0H0
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0;g%.y@$0&D0H00;g%.y\RQ0&FDDG\SSE0H00;
`)&2\g%X}0&(%.`)'-0eM,%4}%&D0H00;g%.y
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0H00;
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!$%D
&I0)4g-%(%.$M-@$?0&D0)4g-%.$M-@$•+-4%.U

4l•MP%K+(
(V •%0&'()%*+0H00;4%.U4l•K+(7
'V &'()%*+0H0;4g4%@$@5.U4l•`W%@5>=%'k10)4g4%
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Mỗi người sử dụng mạng máy tình đều có mật khẩu. Giả sử mỗi mật khẩu gồm 6 kí tư, mỗi kí
tự hoặc là m cột chữ số ( trong 10 chữ số từ 0 đến 9) hoặc là một chữ cái ( trong bảng 26 chữ
cái tiếng Anh) và mật khẩu phải có ít nhất là một chữ số.

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hoặc là một chữ số ( trong 10 chữ số từ 0 đế 9)
b) Có bao nhiêu dãy gồm 6 kí tự nói ở câu a) không phải là mật khẩu?
c) Có thể lập được bao nhiêu mật khẩu?
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Biển số xe máy của tỉnh A ( nếu không kể mã số tỉnh) có 6 kí tự, trong đó kí tự vò trí đầu tiên

là một chữ cái (trong bảng 26 chữ cái tiếng Anh), kí tự ở vò trí thứ 2 là một chữ số thuộc tập
{ }
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Trong một trường THPT, khối 11 có 280 học sinh nam và 325 học sinh nữ.
a) Nhà trường cần chọn một học sinh khối 11 đi dự dạ hội của học sinh thành phố. Hỏi
nhà trường có bao nhiêu cách chọn?
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Gv: Lê Tuấn Cảnh
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!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(6

Gv: Lê Tuấn Cảnh
!$%
&'()%*+0H0,bM|lMD;0,%@$)W%.`)2O.0H%'$<$%430uW%7
89u%0H0,bM|lMD;0,%@$)D0uW%>$)H@‚03(DMh.~
RQ,-0H0,bM|lM>$9

GD‹GD\GD0H0
!$%
&'()%*+0H0,bM|lM0uW%0)Bg%XH0@$)2g%lX*.$2O.<nQ7
89u%0H0,bM|lM0uW%03(BXH0.a)$(0)2O.)H@‚03(B@$J0
>=%RQ0&B‹0H0,bM|lM
!$%\
&.T>RM4J0'()%*+,-W2D01,-XH0(+.U0H001,-??\?D7
89`*.RM5
{ }

CC\CDB =
:;%,-0h.i20&<=
abcd
T.$>RM,-W2'-01,-
4&.(0h|lMD01,-03(.RM5!@$)D@‚.`s$i(?$.`{2'?$0N00@$$
4@‚<RQ0&.€.0P9
D
GD‹GD,-.)PQ0'.^8dK+Q.b04l24T%P%'$%$QV
!$%D
&.T>RM4J0'()%*+01,->YW2{201,-XH0(+.U.RM
{ }
BCCC\CDB =
7
89:;%,-0h.i20&<=
{ }
C BC C\abcde a e≠ ∈
(|f.(%.`gJM9
8=,-9
C Babcd a ≠
;
{ }
C\CDa ∈
0&\0H00;?0;
{ } { }
? ? BCC\CD [b c d a∈
.i,-0H00;>$,-0H0,bM|lM'(,-.+x]03(.RM
{ } { }
BCC\CD [ a
@$)i'?$.`{20@$$0N0<#*0&
\

G\‹GE0H0
RQ0&9\EGF,-<=
abcd
?8=,-
\C Babcd a ≠
•s>+R../.(0&F,-<=
\abcd
&2>=%?.(0&9FmFG\E,-.)PQ0'.
!$%I
`)2O.@v>)=%†>Q2M%0?.`*.H24g'%?F@R4O@%*X}0d2O.>t0@54s0
•%0&'()%*+0H0,bM|lM=|PQ`(7
89€.0PF@R4O@%*45+@54s0X}0d2O.>t0^X}(%4l4s00d@A%
2O.g%XH0V.`*F4g'%?.i0H0,bM|lM=F@R4O@%*>$2O.)H@‚03(F
Mh.~X%,bM|lM@$)F@‚.`s^.y=VMc'%L.?X}>eM
#*.(0&9
F
GF‹GDB\BXl.K+P
!$%E
s.Ž03(.€.0P0H0,-W2D01,-XH0(+@$,-4n>RM4J0.U5
{ }
CC\CDB =
'k
MfM)H@‚7
89fM)H@‚.`*5!0).(.$>RM0H0,-W2'-,-XH0(+>$9
D
GD‹GD,-
T]`k?.€.0P0H0,-45+@%l.<A%<=0eM4}%,(+9
\D D\ D\ D\ D\ \D \D \D \D D\ D\ D\
C C C C C C C C C C C
D\ D\ D\ D\ D\ \D \D \D D\ \D \D \D

           
           
           
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RQ.ŽGIIIIGEEEEB
!$%S
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(7

Gv: Lê Tuấn Cảnh
y2%`k.`*.RM
{ }
CC\CDCICECSB =
0&.T>RM.$4J00H0,-W2'PQ01,-XH0
(+2$.Ž03(0t.i0%(l.0)SB
89fM)H@‚
S
GS‹GIBDB?0).(,-0H0,-W2S01,-XH0(+.$>RM4J0.U!
T]`k.`)IBDB,-.i24J0?.(>+}@%l.4J09
IBDB
IB

=
0eM,-0&.Ž>$FFFFFFF
#
\DIES \DIES \DIES
C C C
SEID\ ESID\ ISED\
  
  

  
#*.Ž03(0t>$9GIBFFFFFFF
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IB 9wB F
FFFFFFF9F 
=


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
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!$%F
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,()0)9
(V !=W%0s%1(7
'V (%'=@$•W%_(%4h+l7
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D
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^0H0VRQ0&.€.0P>$D0H0|lM.)PQ0'.
'VlM?•W%_(%4h+l0&‹G^0H0V?|lM!??8@$)'(0u0v>=%0&\‹GE
^0H0VRQ0&.€.0P>$EG0H0.)PQ0'.
!$%w
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,%W2I(2@$I1•%0&'()%*+0H0,bM|lM0uW%?l+9
(V €.0P0H0;0,%W%.+x]7
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89(V(%0H%'$@$Bl?*X%|lMB;0,%W%.+x]?4&>$)H@‚03(B;0,%
y@A%BlRQ0&

B
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0H0|lMRQ(%0H%'$0&9^I‹V^I‹VGFFBB0H0|lM.)PQ0'.
!$%B
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89RM5
{ }
BCC\CECwB =
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D
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D‹,-@RQ
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>)=%'•4%0H0)H@‚4h+>$B(0&9
D
GD‹`)4&
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q
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Gv: Lê Tuấn Cảnh
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{ }
CC\CDCICSCwB =
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RQ,-0h.i2>$9
S
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RQ,-.)PQ0'.9S‹q‹E‹G\EBB^,-V
!$%\
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RQ,-0H0,bM|lM>$9
E
GE‹GSB^0H0V
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I
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………………………………………………………………………………………………………………………………………………………………………………………………………………
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@A%4%T24h+@$4%T20+-%
>$0H04z03(4(%H07
89(%H0>W%0&I0=*0&I4z?(%4z.i>+}Mc'%L.(+@$0y\4z.i
X}.•$8)4&.(>€Q4%T2.+x].`)I4%T2.i,-@a0.>RM4J0>$2O.0zJM
0RM03(IMh.~RQ,-@a0.>$9

I
I‹
ID B
^I V‹
A = = =
−
^@a0.V
!$%\

O.0c+>=0'O)H;0>t0.$>RM0&D.$@%*?0h'h+0;`(2O..$@%*>$2s(2
4-0•!?2O..$@%*>$2M&%H24-0•!@$2O..$@%*>$2Xl.)H.`_•!•%
0&'()%*+0H00;4T'h+2$X}0&(%X%*2%L27
89j%'h+0;\.$@%*.`)D.$@%*`(>$2%H24-0?M&%$24-0@$Xl.)H
.`_^X‘V.i.y./0h4P2'P)#*0H0,-0H00;4T+'h+g%X}X%*2%L2
>$9

\
D
D‹
FD
^D \V‹
A = =
−
^0H0V
!$%D&'()%*+,-+Q*<W2I01,-XH0X}@$XH0(+4}%2O.7
89u%,-0h.i20&<=9
  \ D I
a a a a a
?.`)4&
C
i j
a a i j≠ ≠
@$
{ }
CC\CDCICECSCFCw ? ??I
i
a i∈ =
#@RQ.(0&.T0)%2u%,-<=.`*>$2O.0zJM
0RMI03(w01,-RQ,-0h.i2>$9
I
w
w‹
IB
^w IV‹
A = =
−

^,-V
!$%I
:%P,~0&'PQ'})(2$+XH0(+@$')>;XH0(+•%0&'()%*+0H00b2'('}
)(@P)'(>;4n0)^2u%>;0b22O.'}V7
89i'PQ'})(2$+XH0(+@$'(>;0b2)(XH0(+*2u%>h0;`('('}
)(4T0b2@$)'(>;?.(0&2O.0zJM0RM\03(SMh.~RQ,-0H00b2)(@$)'(>;
XH0(+>$9
\
S
S‹
B
^S \V‹
A = =
−
^0H0V
!$%E
&'()%*+0H02b0-%.%lMD'&4’4J00;.UE'&4’X(0(+7
89b0-%.%lMD'&45.UE'&4’XH0(+>$2O.0zJM0RMD03(EMh.~
RQ,-0H02b0>$9
D
E
E‹
\EB
^E DV‹
A = =
−
^0H0V
!$%S
U5
{ }

BCC\CICSB =
0&.T>RM4J0'()%*+,-W2'(01,-XH0(+7
89:;%,-0h.i20&<=9
C Babc a ≠
@$|f.(%.gJM
;
{ }
[ Ba B∈
GZ0&D0H00;
;
{ }
? [b c B a∈
.4@%L0,bM|lM01,-.+x]03(
{ }
? [b c B a∈
@$)(%
@‚.`s0v>=%^X‘@$.i.y./MP%4P2'P)VGZ0&

D
D‹

^D V‹
A = =
−
0H00;
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(10

Gv: Lê Tuấn Cảnh
RQ,-0h.i2>$9DGDF^,-V
H0XH09-0&Œ(@$X}0&Œ(W2'(01,->RM4J0.U!>$2O.0%JM

0RM\03(IMh.~.`)!
\
I
I‹
EB
^I \V‹
A = =
−
^,-V-0H0,-Œ(9
Bbc
0h>)=%'•4%
.4@%L0,bM|lM
{ }
? C\CICSb c ∈
@$)(%@‚.`s0v>=%@$.i.y./MP%'P)4P2
-4&>$0zJM0RM03(DMh.~9

D
D‹

^D V‹
A = =
−
^,-V
RQ,-0h.i2>$9EBqGDF,-
!$%F
).RM5
{ }
BCCC\CDCIB =
&.T>RM4J0'()%*+,-06?2u%,-W2I01,-XH0

(+7
89:;%,-0h.i2>$9
{ }
C BC BCCDabcde a e≠ ∈
@$
{ }
? ? ? BCCC\CDCIa b c d ∈
f.0H0.`gJM9
8=,-
BC Babcd a ≠
?;
{ }
? ? ? CC\CDCIa b c d ∈
0&
D
I
I‹
B
^I DV‹
A = =
−
^,-<=
Babcd
V
8=,-
C DC Babcd abcd a ≠
;
{ } { }
C\CDCI   CC\CIa hay a∈ ∈
45+0&D0H0

0;?0;
{ }
? ? ? CC\CDCIa b c d ∈
0&
\
D
D‹
D
^D \V‹
A = =
−
,-RQ,-<=
C DC Babcd abcd a ≠
0&DDGw^,-V
RQ,-0h.i2>$9BmwG\^,-V
!$%w
A%.RM5
{ }
BCCC\CDCICEB =
?.(0&.T>RM4J0'()%*+,-W2I01,-XH0(+@$
.`)4&MP%0&2e.01,-I7
89:;%,-0h.i20&<=9
C BC ? ? ? ?abcde a a b c d e B≠ ∈
-0&I01,-MP%0&2e.01,-I.(
|f.0H0.`gJM9
8=
Ibcde
?0;
{ }
? ? ? BCCC\CDCEb c d e∈

0&
D
E
E‹
\EB
^E DV‹
A = =
−
^,-V
8=0H0
( )
I I C I C I C Ba cde ab de abc e abcd a ≠
;
{ }
CC\CDCEa ∈
0&I0H00;?
0;
{ } { }
? ? BCCC\CDCE [b c d a∈
0&
\
I
I‹
EB
^I \V‹
A = =
−
^,-V
&'-,-<=.`**0&DEBGBB^,-V
RQ0&\EBmBBGIEB,-.)PQ0'.

!$%B
A%0H001,-CC\CDCI0&.T>RM4J0'()%*+,-9
(V •$,-060&'(01,-XH0(+7
'V :W2'(01,-XH0(+@$X}>A\DI7
0V •$,-060&'(01,-XH0(+@$X}>A\DI7
89(V:;%,-0h.i2W2'(01,-0)H<=
abc
@$>$,-069;
{ }
CDc ∈
0&0H0
0;?0;
{ } { }
? CC\CDCI [a b c∈
0&

D
D‹

^D V‹
A = =
−
0H00;
RQ,-0h.i2>$9GD^,-V
'V-0h.i20&<=
\DIabc ≤
(|f.0H0XP{,(+9
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(11

Gv: Lê Tuấn Cảnh

j%
 \DIbc ≤
•t04&0&.T.+x]0;
{ }
? C\CDCIb c ∈
?.(0&

D
D‹

^D V‹
A = =
−

^,-V
j%
 \DIbc ≤
•t04&0&.T.+x]0;
{ }
? C\CDCIb c ∈
?.(0&

D
D‹

^D V‹
A = =
−

^,-V

\j%
\ \DIbc ≤
•t04&.(0;
{ }
CCDb ∈
0&\0H00;?0;
{ } { }
CCDCI [c b∈
0&\0H00;RQ0&\\Gw^,-V
&2>=%?0&9mmwG\\,-.)PQ0'.
0V-0h.i20&<=
\DIabc ≤
(|f.0H0XP{,(+9
j%
 \DIbc ≤
;
{ }
CDc ∈
0&0H00;?0;
{ } { }
C\CDCI [b c∈
0&\0H0
0;(0&\GE,-<=
 \DIbc ≤
j%
 \DIbc ≤
;
{ }
Dc ∈
0&0H00;?0;

{ }
C\CIb ∈
0&\0H00;
(0&\G\,-<=
 \DIbc ≤

\j%
\  \DIb ≤
;
{ }
CDb ∈
0&0H00;(0&,-<=
\  \DIb ≤
\j%
\ D \DIb ≤
;
{ }
Cb ∈
0&0H00;(0&,-<=
\  \DIb ≤
&2>=%0&9Em\mmG\,-.)PQ0'.
!$%
US01,-BCCC\CDCICE0&.T>RM4J0'()%*+,-06?2u%,-W2I01,-XH0(+7
89-0h.i20&<=
{ }
C BC ? ? ? ? BCCC\CDCICEabcde a a b c d e B≠ ∈ =
@$>$,-06
8=
Babcd
;

{ }
? ? ? CC\CDCICEa b c d ∈
0&
D
E
E‹
\EB
^E DV‹
A = =
−
,-<=
Babcd
8=0H0
( )
 DC E C Babcd abcd abcd a ≠
;
{ } { }
C\CDCICE [a e∈
0&I0H00;?
0;
{ } { }
? ? BCC\CDCICE [ Cb c d a e∈
0&
\
I
I‹
EB
^I \V‹
A = =
−

^,-VRQ0&IEBG\BB,-<=
abcd
&'(,-<=.`**0&9\\BBGwBB,-
&2>=%0&9\EBmwBBGEB,-.)PQ0'.
!$%
&'()%*+,-./%*W2I01,-XH0(+0%(l.0)B^01,-$@=XH0BV7
89-0&I01,-XH0(+0%(l.0)B0&<=9
BC Babcd a ≠
.`)4&
{ }
? ? ? CC\CDCICECSCFCwa b c d B∈ =
<)
Ba
≠
?X%4&.(0&
D
w
w‹
\BD
^w DV‹
A = =
−
,-.)PQ0'.
!$%\
)E01,-CC\CDCICE&.T.=)`('()%*+,-W2D01,-XH0(+7`)4&0&'()
%*+,-0%(l.0)I7
89-W2'-01,-XH0(+0&<=
C Babcd a ≠
.`)4&
{ }

? ? ? CC\CDCICEa b c d B∈ =
*
.(0&9
D
E
E‹
\EB
^E DV‹
A = =
−
^,-V
-
C Babcd a ≠
0%(l.0)IX%<GI@$0;
{ }
? ? CC\CDCEa b c ∈
0&
\
I
I‹
EB
^I \V‹
A = =
−
^,-V
!$%D
U.RM5
{ }
BCCC\CDCICEB =
0&.T>RM4J09

(V !()%*+,-./%*W2I01,-XH0(+7
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(12

Gv: Lê Tuấn Cảnh
'V !()%*+,-./%*06W2I01,-XH0(+7
89(V#l+XT0P.`gJM,-B4y4h+?.i.(0&9
I
S
A
,-./%*W2I01,-XH0(+
`)
I
S
A
0H0,-4&W20&
D
E
A
,-W2I01,-2$01,-B4y4h+RQ,-W2I
01,-XH0(+>RM.U.RM5!>$9
I D
S E
EBA A− =
^,-V
'Va2'$%
!$%I
f.0H001,-W2w01,-?.`)4&0&I01,-@$D01,-0v>=%>$?\?D?I•%0&'()
%*+,-.l?l+9
(V I01,-4J0|lMX5(+7
'V H001,-4J0|lM.+x]7

89(V:;%&2>$,-(!$%.)HQ*+0h+.(0h,bM|lM{2,-9(??\?D?I@$)I@‚.`s
XH0(+-0H0,bM|lM>$9
I
GI‹GB,-.)PQ0'.
'V•RM2O.,-0&w01,-.)P2nQ*+0h+?./00€.>$@%L0|lM'-,-?\?D?I@$)D@‚.`s
.+x].`)w@‚.`s?0vI@‚.`s0v>=%.i01,->eMI>h
RQ0&9
D
w
w‹
\BD
^w DV‹
A = =
−
,-.)PQ0'.
!$%E
).RM5
{ }
BCCC\CDCICECSX =
&.T>RM4J0'()%*+,-W2I01,-XH0(+4}%
2O..U^01,-4h+.%*M$%XH0BV.`)2u%.`gJM,(+9
(V >$,-067
'V O..`)'(,-4h+.%*MP%'k7
89f..RM5
{ }
BCCC\CDCICECSX =
(V 8=,-0h.i2>$
abcde
^XT0P(GBV@$>$,-06GZ0&2


GD0H00;
{ }
BCCDCEe ∈
?0;
{ }
? ? ? [a b c d X e∈
0&2

G
D
S
FDBA =
0H0RQ0&2

2

G\\EB,-
06^XT0P
Bbcde
V(>)=%'•,-0&<=
Bbcde
&2
\
G\0;
{ }
CDCEe ∈
@$0&2
D
G
\

E
BA =
0H00;
{ }
? ? [ BCb c d X e∈
RQ0&2
\
2
D
G\EB,-06<=
Bbcde

&2>=%?0&\\EBq\EBG\BBB,-.)PQ0'.
'V./|f.9G
abcde
.`)0H0XP{,(+9
a20H0,-
abcde
^XT0P(GBV0&2

G\0H00;@‚.`s0)^>$()e0>$')e0>$0V
(+4&0;@‚.`sXH0(+0)D@‚.`s0v>=%.U
{ }
[ X
,-0H00;92

G
D
S
FDBA =

0H0
@RQ0&9\FDBGIB,-^XT0P
Bbcde
V.)PQ0'.
f.,-
Bbcde
0&0H00;@‚.`s0)^.=%')e0.=%0V(+4&0;@‚.`sXH0(+0)\
@‚.`s.y0v>=%.U
{ }
[ BCX
?,-0H00;>$
\
E
BA =
0H0“Q0&BGDB,-
<=
Bbcde
&2>=%0&9IBqDBGFB,-.)PQ0'.
…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
 
 #9).RM0&Mh.~@$,-+Q*X@A%
 k n≤ ≤
u%.RM0)03(0&XMh.~
4J0;%>$2O..ŽJM0RMX03(Mh.~03(^;%.b.>$2O..ŽJM0RMX03(V
!$%.RM=%,-@$:%P%.s0$%>%*+>+$O%'O`(13

Gv: Lê Tuấn Cảnh
 -.ŽJM0RMX03(Mh.~9js%L+
^ ? ŠV
k

n
C k n n≤ ≤ ∈¥
?
‹
‹^ V‹
k
n
n
C
k n k
=
−
(Q
‹ ^ V^ V^ V
‹^ V‹ ‹ ‹
k
k
n
n
A
n n n n n k
C
k n k k k
− − − +
= = =
−
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\
B
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B
\‹^B \V‹
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B
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0f)0h.i2>$9
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Gv: Lê Tuấn Cảnh
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PHẦN BÀI TẬP TRẮC NGHIỆM
Chương II. TỔ HP – XÁC XUẤT
Câu 1. Với bốn chữ số 1, 2, 3, 4 có thể lập được bao nhiêu số có các chữ số phân biệt ?
A. 16 B. 24 C. 32 D. 64
Câu 2. Với các chữ số 0; 1; 2; 3; 4; 5; 6 có thể lập được bao nhiêu số chẵn gốm 5 chữ số đôi
một khác nhau ( chữ số đầu tiên phải khác 0) ?
A. 1250 B. 1260 C. 1280 D. 1270
Câu 3. Với 10 chữ số 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 có thể lập được bao nhiêu số có 5 chữ số phân
biệt?
A. 27216 B. 7216 C. 2216 D. 2716
Câu 4. Cho 5 chữ số 0, 1, 2, 3, 4. Từ 5 chữ số đó có thể lập được bao nhiêu chữ số chẵn có
năm chữ số sao cho trong mỗi chữ số đó mỗi chữ số trên có mặt một lần ?
A. 24 B. 36 C. 60 D. 82
Câu 5. Có bao nhiêu cách chọn một tập hợp 5 chữ từ bảng chữ cái tiếng Anh ?
A. 65780 B. 56780 C. 57680 D. 57860
Câu 6. Một bài trắc nghiệm khách quan có 10 câu hỏi. Mỗi câu hỏi có 4 phương án trả lời. Số
phương án trả lời là:
A. 10
4
B. 4
10
C. 40 D. 4
Câu 7. Số các số tự nhiên có 6 chữ số và chia hết cho 5 bằng:
A. 17280 B. 30 C. 180000 D. 86400
Câu 8. Số các số tự nhiên có 3 chữ số khác nhau đôi một và khác 0 mà tổng các chữ số của
chúng bằng 8 là:

A. 6 B. 12 C. 24 D. 36
Câu 9. Ta xếp 5 quả cầu trắng(khác nhau) và 5 quả cầu xanh (khác nhau) vào 10 vò trí theo
một dãy, sao cho các quả cầu cùng màu không được cạnh nhau. Số cách xếp bằng:
A. 12! B. 14000 C. 240 D. 28000
Câu 10. Thành lập từ các chữ số 0,1,2,3,4,5. Số các số tự nhiên chẵn có 4 chữ khác nhau đôi
một khác nhau là:
A. 156 B. 144 C. 128 D. 516
Câu 11. Lấy hai con bài từ cỗ bài tú lơ khơ 52 con. Số cách lấy là:
A. 104 B. 1326 C. 450 D. 2652
Câu 12. Năm người xếp vào ngồi quanh một bàn tròn năm ghế. Số cách xếp là:
A. 50 B. 100 C. 120 D. 24
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