510.76
rS. VO THE HirU - NGUYEN VINH CAN
H419V
•
•
HOC a ON LUYIN
T H E O C A U T R U C D E THI
MON
I
TS. VU THE HlfU - NGUYEN VINH CAN
nioc & ON LUYEN
T H E O C A U T R U C D E THI
THi; VIEN TINH BINH THU*N
ON THI
DAI HOC
Ha
NQI
N H A X U A T B A N D A I HQC QUOC G I A H A N O I
H O C vA
NHA XUAT B A N DAI HQC QUOC GIA HA NQI
16 Hang Chuoi - Hai Ba TrUng - Ha Npi
Dien thoai: Bien tap - Ciie ban: (04) 39714896;
Hanh ctiinii: (04)3 9714899; Tong Bien tap: (04) 39714897
Fax: (04) 39714899
Chiu trdch nhiem xuat ban:
Gidm doc - Tong bien tap: T S . P H A M T H I T R A M
Nha sach H O N G A N
Che ban:
THAI VAN
Sica bai:
H O N G SON
Bien tap:
Trinh
bay bia:
THAI HOC
Thj^c hi?n lien kit: Nha s a c h H O N G A N
SACH LIEN KET
O NLUYEN THEO CAU TRUC D ET H I M O N TOAN THPT
IVla so: 1L - 65DH2013
In 2.000 cuon, I
1). M 5 i each sap thuf til eac
phan tiif ciia t a p hop A duoc goi l a m o t hoan vi ciia n p h a n tuf eua A .
Dinh li : So hoan v i khac nhau ciia n p h a n til bang :
Pn = n ( n - l ) ( n - 2 ) . . . 2 . 1 = n !
3.
C h i n h hrfp
M o t tap hop A hOTu h a n gom n phan tuf (n > 1) va so nguyen k
(0 < k < n). M o i tap hop eon eua A gom k phan til sSp theo mot thuf tiT
nhat d i n h dLfgrc goi la mot chinh hap chap k cua n phan tuf.
Dinh li : So c h i n h hop chap k ciia n p h a n tuf bang :
A;; = n ( n - l ) ( n - 2)...(n - k + 1) =
(n - k ) !
4.
(Quy irde : 0! = 1).
T o hofp
Cho t a p h o p A hufu h a n c6 n p h a n tuf ( n > 1) va so nguyen k
(0 < k < n). M o i tap hop con gom k p h a n tuf ciia A (khong t i n h thuf tU
eac p h a n tuf) g o i l a m o t to hop chap k cua n p h a n tuf.
n'
A''
Dinh li : So to hop chap k cua n p h a n tuf l a : C'' =
= —(n-k)!k!
k!
He qua:
Cl=C:=l;
0^= Cr''; C ^ = C!; + C ^ \
HQC va on luy$n theo CTDT mon Toan THPT S 5
1.
a)
BAI TAP
C h o cac chuf so 2 , 3, 4 , 5, 6, 7.
C o b a o n h i e u so' t\i n h i e n c6 h a i chuf so ducfc t a o n e n til t a p h o p c a c
chuf so d a cho.
b)
C o bao n h i e u so t i i n h i e n c6 h a i chuf so k h a c n h a u difOc tao n e n tiT t a p
hcrp chuf so d a cho.
CHI
a)
DAN
D e t a o m o t so c6 h a i chuf so t a t h i f c h i e n h a i c o n g d o a n :
1. C h o n m o t chuf so l a m chuf so h a n g chuc : c6 6 k e t qua c6 t h e .
2.
C h o n m o t chuf so l a m chuf so h a n g d o n v i : c6 6 k e t qua c6 t h e .
T h e o q u y t ^ c n h a n so k e t q u a t a o t h a n h cac so c6 h a i chuf so tii t a p
hgfp 6 chCif so d a cho l a : n = 6 x 6 = 3 6 so'.
b)
L a p l u a n g i o n g n h i f c a u a ) n h u i i g liTu y sir k h a c b i e t so vdfi trifofng h o p
t r e n of cho so d u g c t a o t h a n h c6 h a i chuf so k h a c n h a u . D o do t a c6 k e t
q u a n h u sau :
1. C h o n m o t chuf so l a m chuf so' h a n g chuc : c6 6 k e t q u a c6 t h e .
2.
C h o n m o t chuf so l a m chuf so h a n g d o n v i : c6 5 k e t q u a c6 t h e ( v i chuf
so n a y p h a i k h a c chuf so h a n g chuc d a c h o n trifdrc do).
T h e o q u y t a c n h a n : so cac so c6 h a i chuf so k h a c n h a u difcfc t a o t h a n h
tCr t a p h o p 6 chuf so d a cho l a : n ' = 6 x 5 = 3 0 so.
Cdch
khac
: M o i so c6 h a i chuf so t a o t h a n h tiT 6 chuT so d a cho l a m o t
t a p hop c o n s^p thuf tiT g o m h a i p h a n tuf tiT 6 p h a n tuf d a cho. D o do so
cac so C O h a i chuf so k h a c n h a u t a o t h a n h tiT 6 chuf so d a cho l a so
c h i n h hop c h a p 2 cua t a p hop 6 p h a n tuf.
n = Ag = 6.5 = 30 so.
2.
C h o t a p h o p cac cha so 0, 1, 2, 3 , 4, 5, 6.
C o bao n h i e u so t U n h i e n c6 4 chuf so k h a c n h a u tCmg d o i tii t a p
b)
C o bao n h i e u so t u n h i e n c6 4 chuf so tii t a p h o p cac chuf so d a cho.
a)
hop
cac chuf so d a cho.
CHI
a)
DAN
Co 6 each c h o n chuf so h a n g n g h i n (chuf so d a u t i e n p h a i k h a c 0), 7 e a c h
c h o n chff so h a n g t r a m , 7 e a c h c h o n chuf so h a n g chuc v a 7 e a c h c h o n
chuf so h a n g
d o n v i . T h e o q u y tSe n h a n
: so each t a o t h a n h so t i f
n h i e n 4 chuf so tii t a p h o p 7 chiJ so d a cho l a :
N = 6 x 7 x 7 x 7
b)
= 2 0 5 8 so.
C o 6 e a c h c h o n chuf so h a n g n g h i n , k h i c h o n x o n g chuf so h a n g
nghin
c o n l a i 6 chuf so k h a c vdi chuf so h a n g n g h i n d a c h o n . V a y c6 6 e a c h
chon
chuf so h a n g
t r a m . K h i d a c h o n chuf so h a n g
nghin va hang
t r a m , e o n l a i 5 ehOf so k h a c v d i cac chuf so d a c h o n . D o do eo 5 e a c h
6 :S; IS. Vu The Hi/u - NguySn Vinh Cin
chpn chOf so h a n g chuc. Tifcfng tir, c6 4 each chon chOr so h a n g don v i .
Theo quy tac n h a n . So cac so txi n h i e n c6 4 chOf so khac nhau tCfng doi
difcfc tao t h a n h tix t a p hop 7 chuf so da cho l a :
N ' = 6 X 6 X 5 X 4 = 720 so.
Cdch lap luan khdc : M o i so tiT n h i e n c6 4 chOf so khac nhau tao
t h a n h tCr tap hop 7 chOf so da cho l a m o t c h i n h hgrp chap 4 ti^ t a p hgfp
7 chuf so m a cac c h i n h hgfp nay k h o n g c6 chuT so 0 or dau. Do do so cac
so CO 4 chijf so khac nhau tiT 7 chijf so l a :
N' =
- Ag = 7 X 6 X 5 X 4 - 6 X 5 X 4 = 720 so.
3.
Mot to hoc sinh c6 10 ngUofi xep thijf tif thanh hang 1 de vao lorp. Hoi
a) Co bao n h i e u each de to xep h a n g vao l(Jp.
b) Co bao n h i e u each de to xep h a n g vao Idfp sao cho h a i b a n A v a B eua
to luon d i canh nhau va A dufng tri/dtc B .
CHI
D A N
a) So each xep h a n g bang so hoan v i ciia 10 p h a n tiif.
N i = 10! = 3628800 each.
b) Coi h a i b a n A va B n h i i m o t ngudi. Do do so each xep h a n g ciia to de
vao 16p t r o n g do h a i b a n A v a B d i l i e n nhau bang so hoan v i cua 9
phan tijf.
N2 = 9! = 362880 each.
4.
Co bao nhieu each xep 6 ngiicfi ngoi vao m o t ban a n 6 cho t r o n g cac
triiofng hcfp sau :
a) S^p 6 ngiTofi theo h a n g ngang ciia m o t ban a n d a i .
b) S4J) 6 ngiTori ngoi vong quanh m o t b a n a n t r o n .
CHI
D A N
a) M o i each ngoi theo h a n g ngang l a m o t hoan v i cua 6 p h a n tijf. So' each
sap xep l a : 6! = 720 each.
b) Gia suf 6 ngifofi a n diTOc d a n h so thijf t i f la : 1, 2, 3, 4, 5, 6 v a m o t each
sap xep theo b a n t r o n n h i i h i n h .
2 5 1 3 6 4
(1)
5
1 3 6 4 2
(2)
1 3 6 4 2 5
(3)
3 6 4 2 5 1
(4)
6 4 2 5 1 3
(5)
4 2 5 1 3 6
(6)
Neu t a eat b a n t r o n a v i t r i giCfa 2 va 4 r o i t r a i d a i theo b a n ngang
t h i t a CO hoan v i (1) tUofng ijfng m o t each xep ngiiofi ngoi theo ban a n
dai. TiTofng txi c a t of v i t r i giufa 5 va 2. N h u vay m o t each sap xep theo
ban t r o n tiiOng ufng vdri 6 each s a p xep theo b a n d a i . Do do so each
Hoc va on luyen theo CTDT mon Toan THPT
7
5.
x e p 6 ngiTofi n g o i q u a n h b a n a n t r o n l a :
N = — = 120 e a c h .
6
M o t t o CO 15 ngifofi g o m 9 n a m v a 6 nOf. C a n l a p n h o m cong t a e eo 4
ngUdri. H o i eo b a o n h i e u e a c h t h a n h l a p n h o m t r o n g m o i trifcfng h o p
sau d a y :
a) N h o m c6 3 n a m v a 1 nur.
b ) So n a m v a nCf t r o n g n h o m b a n g n h a u .
c) P h a i CO i t n h a t m o t n a m .
CHI
DAN
9 8.7
a) So e a c h c h o n 3 n a m t r o n g so 9 n a m l a : Cj! = "
= 84
1.2.3
So e a c h c h o n 1 niJ t r o n g so 6 nOr l a : Cg = 6
So each t h a n h l a p n h o m g o m 3 n a m v a 1 nCf (theo quy tSc n h a n ) l a :
N i = C^C^ = 5 0 4 e a c h .
b ) So e a c h l a p n h o m g o m 2 n a m v a 2 nuf l a :
N2=
C^C^ = —
'
'
1.2
1.2
= 540 each.
c) So e a c h t h a n h l a p n h o m 4 ngu'ofi t r o n g do c6 i t n h a t 1 n a m l a : 1
nam,
3 nO h o a c 2 n a m , 2 nuf hoae 3 n a m , 1 nur h o a c 4 n a m .
— Cg.Cg +
.Cg 4~ Cg.Cg "I" Cg
_ 6.5.4
9.8 6.5.
9.8.7 ^
9.8.7.6
= 9.
+
+
.6 +
= 1350 each.
1.2.3
1.2 1.2
1.2.3
1.2.3.4
Ghi chu : C u n g eo t h e l a p l u a n n h i f sau :
C a t o CO 15 ngiTcfi. So e a c h l a p n h o m 4 n g U d i t u y y l a :
^ 4 ^ ^ — 1^4 .x1 3o. 1^2 ^ ^ g g g ^ ^ ^ ^
_
1
5
.
~
1.2.3.4
1.2.3.4
So e a c h l a p n h o m 4 ngUofi t o a n nuf l a : C ^ = Cg = 6.5. = 15 e a c h .
1.2
So e a c h l a p n h o m 4 n g i / d i eo i t n h a t 1 n a m l a :
N = CJs - C^ = 1365 - 15 = 1 3 5 0 e a c h .
T r o n g m a t p h a n g eo n d i e m p h a n b i e t ( n > 3 ) t r o n g do eo d i i n g k
6.
d i e m n S m t r e n m o t d i i d n g t h S n g (3 < k < n ) . H o i c6 bao n h i e u t a m
g i a c n h a n cac d i e m d a cho l a d i n h .
CHIDAN
Cuf 3 d i e m k h o n g t h S n g h a n g t a o t h a n h m o t t a m g i a c . So cac t a p h o p
c o n 3 d i e m t r o n g n d i e m l a : C^. So cac t a p c o n 3 d i e m t r o n g k d i e m
t r e n diTcfng t h i n g l a : C^. So t a m g i a c c6 3 d i n h l a cac d i e m d a cho
l a : N = C^ - Cl t a m g i a c .
8 ;
TS. Vu The Hi/u - Nguyen VTnh Can
7. a) Co b a o nhieu so t i i n h i e n l a so chan c6 6 chiif so doi m o t khac nhau
va chuf so dau t i e n la chOf so le.
b) Co bao nhieu so t i i n h i e n c6 6 chuf so doi mot khac nhau, trong do c6
dung 3 chuf so le, 3 chuf so chSn (chuf so dau t i e n phai khac 0).
CHI D A N
a) So can t i m c6 d a n g : x = a^agaga^agag t r o n g do a i ,
ae
l a y cac chOf
so 0, 1, 2,
8, 9 vdfi a i ?i 0, ai aj v d i 1 < i ?i j < 6.
- V i X la so chSn nen ae c6 5 each chon tiT cac chuT so 0, 2, 4, 6, 8.
- V i a i la chuT so le nen c6 5 each chon tiT cac chuf so 1, 3, 5, 7, 9.
Con l a i a2a3a4a5 l a m o t chinh hop chap 4 eiia 8 chuf so con l a i s a u k h i
da chon ae va a i . Theo q u y t^c n h a n , so cac so can xac d i n h l a :
N i = S.S-Ag = 5.5.8.7.6.5 = 42000 so.
b) M o t so theo yeu c a u de b a i gom 3 chuf so tii tap X i = |0; 2; 4; 6; 81 va
3 chuf so tCr t a p hop X2 = I I ; 3; 5; 7; 91 ghep l a i va loai d i cac day 6
chuf so CO chuf so 0 dufng dau.
So each lay 3 chuf so thuoc t a p X i la : Ci? = 10 each.
So each lay 3 p h a n tuf thuoc X2 l a : Cg = 10 each.
So' each ghep 3 p h a n tuf l a y txi X i v o i 3 p h a n tuf l a y tii X2 l a :
C^C^ = 10.10 = 100 each.
So' day so' eo thuf t i f eiia 6 p h a n tuf diioc ghep l a i l a :
100.6! = 72000 day.
Cac day so c6 chuf so 0 a dau g o m 2 chiJ so khac 0 ciia X i va 3 chuf so'
ciia X2 : So cac day so nhif t r e n la : C 4 . C 5 . 5 ! = 7200 day.
So cac so theo yeu cau de b a i la :
N2 - C ^ C ^ 6 ! - C ^ C ^ 5 ! = 72000 - 7200 = 64800 so.
8.
M o t hop diing 4 v i e n b i do, 5 v i e n h i t r a n g va 6 v i e n b i vang. NgLfofi
ta chon r a 4 v i e n b i t i f hop do. H o i c6 bao nhieu each l a y de t r o n g so
b i j a y r a k h o n g dii ca 3 mau.
CHI D A N
Cdch 1 : So each chon 4 v i e n b i k h o n g d u 3 mau b a n g so' each chon 4
v i e n b a t k i trir d i so each chon 4 v i e n c6 ca 3 mau.
N = Cjg - (C^ .C^ .C^ + C^ .C\ + Cl .C\) = 645 each.
Cdch 2 : So each chon 4 v i e n b i k h o n g d u 3 m a u bang so each chon 4
v i e n m o t m a u (4 do, 4 t r a n g va 4 vang) cong v d i so each chon 4 v i e n
hai mau ( 1 do, 3 t r a n g hoae 2 do, 2 t r a n g hoac 3 do, 1 t r a n g hoae 1
do, 3 v a n g hoac 2 do, 2 v a n g hoae 3 do, 1 v a n g hoae 1 t r S n g , 3 v a n g
hoae 2 trSng, 2 v a n g hoac 3 t r a n g , 1 vang).
N = c : +Ct +CI+
ClCl
+ ClCl
+ C^C^ + ... + C^C^ = 645 caeh.
Hoc va on luyen theo C T D T m o n loan T H P T SI 9
9.
Co 15 n a m va 15 nuT k h a c h du l i c h dijfng t h a n h vong t r o n quanh ngon
lijfa t r a i . H o i c6 bao n h i e u each xep de k h o n g eo triTcfng hop hai ngi/6i
eCing gidfi canh nhau.
CHI DAN
ThiTe h i e n sap xep bang each d a n h so 30 cho t r e n di/orng t r o n tii 1
den 30 va cho n a m dufng so le nuT dufng cho so chSn hoac ngi/gc l a i (2
each). Co 15! each sSp n a m dufng trong cae cho so' le (hoac chSn) va
15! each sdp nuf dufng t r o n g cae cho so ch^n (hoac le).
V i diidng t r o n 30 cho nen m o i each sSp xep nao do xoay tua 30 cho
theo dung t r a t tiT do ta cung chi eo mot each sap t r e n dirofng t r o n
(xem b a i so 4).
Do do so each sSp xep theo difcfng t r o n 30 k h a c h du l i c h theo yeu cau
2.(15!)(15!)
, ,
de la : N =
= 14!.15! each.
30
10. Chufng m i n h cae dang thufc :
a)
+
+ ... +
= ——- (1) t r o n g do A^ la c h i n h hop chap 2 eua n.
Ag
A3
A„
n
b)
CHI
C;; = C;;:; + Cl;}^ + ... + Cl:\) t r o n g do C; la to hop chap r ciia n.
DAN
a) V(Ji k e N, k > 2 ta c6 : A', = k ( k - 1)
^ =
=
- '
^
A^
k(k-l)
k-1
k
Thay k = 2, 3,
n vao (*) ta c6 ve t r a i ciia (1) la :
(1
(I
V
f 1
1^
— —
— — —
— —
+... +
[n-1
nj
u
2. l 2 3v
b) Theo t i n h chat eua to hop ta c6 :
=
Cn_3
(*)
+ C^^g
Cong ve vdi ve cae dang thufc t r e n ta
diTOe :
c:;-c::;+c-^3+c::u...+c:-uc:
Do C;; = C;::} = l n e n thay C;: d dang thufc cuoi bori C^:}
ta
dugfc
dang
thufc (2) can chufng m i n h .
11.
Chufng m i n h bat dang thile :
t r o n g do k e N, k < 2000,
C^ooi +
^ C\Z + CfZ
la to hop chap k eua n p h a n tuf.
1 0 t4l TS. Vu The Hi/u - Nguyin Vinh CSn
CHI DAN
V(Ji 0 < k < 1000 t h i
^2001
k+1
2001
k
'^2001 -
2001!
(k + l ) ! ( 2 0 0 0 - k ) !
k +1
k!(2001-k)!
2001!
2001-k
'-^2001 -
'^2001
-
, plOOO _ p l O O l
•• - '-^2001 ~ ^ 2 0 0 1
pk
*-^2001
<1
p k + 1 ^ plOOO
^2001 — ^2001
plOOl
^^2001
-111-k
M a t k h a c , v 6 i 1 0 0 0 < k < 2 0 0 0 t h e o t i n h c h a t ciia t o h o p C ' = C;;'^ t a
- CIZ'^
• ^ 2 0 0 1 ^ ^-^2001 ~" '^2001
' "^2001
< Cir, +CIZ\i 0 < 2000 - k < 1000,
-
'^2001
' ^2001
0 < 2 0 0 1 - k < 1 0 0 0 t h e o p h a n t r e n d a chufng m i n h .
C A C BAI TAP Tl/ GIAI
12.
TCr d i e m A d e n d i e m B n g i / d i t a c6 t h e d i q u a C h o a c d i q u a D v a
k h o n g CO diTcfng d i t h a n g tii C d e n D . Til A d i t h a n g d e n C c6 2 e a c h ,
t i r C d i t h a n g d e n B c6 3 e a c h . TCr A d i t h a n g d e n D c6 3 e a c h tix D d i
t h a n g d e n B eo 4 e a c h .
a) H o i txi A CO b a o n h i e u e a c h d i tdfi B ?
b)
H o i tCr A d e n B r o i til B trdf v e A
A /
\
CO b a o n h i e u e a c h ?
DS
:
3 X^/
a) 18 e a c h
b) (18)^ e a c h .
13.
4
D
TCr 7 chOf so 0, 1 , 2 , 3, 4, 5, 6 eo t h e g h i dirge b a o n h i e u so tiT n h i e n
m o i so' CO 5 chCT so k h a c n h a u tCrng d o i .
DS : 2 1 6 0 so.
14.
a)
b)
C h o t a p h o p cac chOf so X = |0; 1 ; 2 ; 3; 4 ; 5; 61.
D u n g t a p h o p X eo t h e g h i dufcfe bao n h i e u so tiT n h i e n eo 5 chiT so.
D u n g t a p h o p X c6 t h e g h i dirge b a o n h i e u so t i r n h i e n c6 5 chOf so'
k h a c n h a u tCrng d o i .
c) DCing t a p h g p X c6 t h e g h i dugc b a o n h i e u so tiT n h i e n c6 5 ehCt so
k h a c n h a u l a so e h S n .
DS
15.
:
a ) 6.7* so
b ) 6 l 5 . 4 . 3 so
e) A ^ + 15A^ so.
M o t t o h o c s i n h c6 5 n a m , 5 nOf x e p t h a n h m o t h a n g d o c .
a) Co b a o n h i e u e a c h x e p k h a c n h a u .
b) Co bao n h i e u each x e p h a n g sao cho h a i ngircfi dijfng k e n h a u k h a c g i d i .
DS : a ) 1 0 ! e a c h
16.
b ) 2(5!)^ e a c h .
M o t i g p CO 2 5 n a m h o c s i n h v a 2 0 nuf h o c s i n h . C a n c h o n m o t n h o m
c o n g t a c 3 ngiTdi. H o i eo bao n h i e u each c h g n t r o n g m o i t r i r d n g h g p s a u
a) B a h o c s i n h b a t k i eua Idp.
b)
H a i nijf s i n h v a m o t n a m s i n h .
Hpc va on luyen theo CTDT mon Toan THPT I J : 1 1
c)
B a hoc s i n h c6 i t n h a t m o t nuf.
DS
17.
:
a) C;;^ e a c h
b) 25.C^o e a c h
c) C'^^ -Cl,
each.
Co bao n h i e u e a c h p h a n p h o i 7 do v a t cho 3 n g U d i t r o n g cac trUcfng
h o p sau :
M o i n g u d i i t n h a t m o t do v a t v a k h o n g q u a 3 do v a t .
b)
M o t ngUofi n h a n 3 do v a t , eon 2 ngUcfi m o i ngUcfi h a i do v a t .
a)
DS
18.
: a) 3.C^C^ e a c h
M o t to
CO
b) S.CtCl
+ SCl.Cl
each.
9 n a m v a 3 nOf.
Co bao n h i e u e a c h c h i a t o t h a n h 3 n h o m m o i n h o m 4 ngUofi v a t r o n g
b)
Co bao n h i e u e a c h c h o n m o t n h o m 4 ngUcfi t r o n g do eo 1 nijf.
a)
m o i n h o m c6 1 nuf.
DS
: a) 3.C^ e a c h
b) 3.C;;.2C^ = 10080 each.
19. T i m cac so n g u y e n d u o n g x, y t h o a m a n cac d a n g thufe :
6
f)S
20.
:
X
^ ^ ^ " 5 ^
" 2 ^
•
= 8, y = 3.
Co bao n h i e u so t U n h i e n chain c6 4 ehuf so d o i m o t k h a c n h a u .
DS
21.
•.n=
Al+
4.8.8
= 7 6 0 so.
C h o d a g i a c d e u 2 n d i n h AiA2...A2n, n > 2 n o i t i e p t r o n g d u d n g t r o n .
B i e t r a n g so t a m g i a c c6 d i n h l a 3 t r o n g 2 n d i e m t r e n n h i e u g a p
20
I a n so h i n h ehuf n h a t eo d i n h l a 4 t r o n g 2 n d i n h t r e n . T i m so n .
£>S : n = 8.
22.
T i m so t U n h i e n n , b i e t r a n g C" + 2C;, + 4 C ' + ... + 2 " C " = 2 4 3 .
: n = 5.
T r o n g m o t m o n h o c , t h a y g i a o eo 3 0 c a u h o i k h a c n h a u , g o m 5 cau
24.
G i a i b a t p h u o n g t r i n h ( v d i h a i a n n , k G N)
23.
h o i k h o , 10 c a u h o i t r u n g b i n h v a 15 c a u h o i de. T i r 30 cau h o i do c6
t h e l a p dUcfc bao n h i e u de k i e m t r a g o m
5 cau k h a c n h a u sao
cho
t r o n g m 5 i de n h a t t h i e t p h a i eo d u b a l o a i cau h o i ( k h o , t r u n g b i n h ,
de) v a so c a u h o i de k h o n g i t h o n 2.
DS:n=
25.
Cl,ClCl+C',,C\,Cl+C%C\f
= 56785 d l .
C h o t a p hcfp A eo n p h a n tuf ( n > 4). B i e t r S n g so t a p h o p eon eo 4
p h a n tuf eua A g a p 2 0 I a n so t a p hofp c o n c6 2 p h a n tuf ciia A . T i m so'
t U n h i e n k sao cho so t a p h o p eon eo k p h a n tuf eua A l a I d n n h a t .
: n = 18, C^g > C\^'
flS
o
k = 9.
12 ;.'; TS. Vu Th§' Hyu - Nguygn VTnh Can
§2. NHI THlfC NIUTCfN
K I E N THLTC
1.
N h i thufc N i u t c f n
(a + b ) " = Cf,a"b° + Cla"-'h
+ ... + ClJa'^'^b'^ + ... + C > V = Xc;;a"-''b''
k=0
Ydi
2.
q u y vide a, b ^ 0, a" = b° = 1 , C° = 1 .
Tarn giac P a t c a n
Cac h e so' ciia n h i thufc N i u t o r n ufng vdi n = 0, 1 , 2, 3, ... c6 t h e s a p x e p
diidfi d a n g t a r n g i a c dtfofi d a y g o i l a t a r n g i a c P a t c a n .
1
n =0
n =1
1
n =2
1
n =3
2
1
n =4
1
3
4
n =6
1
3
1;
4 : 1
6
5
n =5
1
10
10
20
15
6
5
15
1
6
1
T r o n g m 6 i k h u n g t h e h i e n t i n h c h a t t o n g h a i h e so h a n g t r e n
so h a n g or h a n g diTdfi h a y C^'^ + Cj^ =
bang
Cl;^i.
BAITAE^
26.
T i m cac so h a n g k h o n g chufa x t r o n g k h a i t r i e n n h i thufc N i u t O n ciia
vdfi X > 0.
/X
J
(Trich
de TSDH
kho'i D nam
2004)
CHI D A N
Vdfi
X >
0, t a
1
CO
: \/x = x ^ ;
I
_ i
= x "*
—j=r
%/x
1
f
/X +•
/x;
= (x'^ + x
7-k
_k
7-1
-
C ° x 3 + Cix
1
3 X 4 + c?x
7-2
2
^ X 4
7
+ ... + C)x^ x"-* + ... + C l x ' ^
So h a n g k h o n g chufa x l a so h a n g thuT k + 1 t r o n g k h a i t r i e n sao c h o :
Hoc va on luyen theo CTOT mon Toan THPT ' 1 3
27.
C^'x
x"^ =C^x 3 ^ " * =C,'x*'
tufc la phai c6 :
- - = 0
3k = 4(7 - k) o k = 4
3
4
Vay so hang khong chijfa x trong khai trien la :
= 35.
Tim so hang chinh giijfa cua nhi thufc NiutOn : (x^ - xy)^*.
CHI DAN
KJiai trien nhi thufc (x^ - xy)^^ c6 15 so hang, so hang chinh giiJa la
so hang thuf 8 c6 dang :
C L ( x ^ r " ( - x y ) ^ = -CLx^^xV^ - -3432x^V^
28.
Tim so hang thuf tii cua khai trien nhi thufc
a
b- a
+
b2 ' - „a2
A"
. Biet
a
rang he so ciia so' hang thuf ba cua khai trien do bSng 21.
CHI DAN
Trong cong thufc nhi thufc NiutOn (A + B)" so hang thuf 3 ciia khai
trien c6 he so la : C? = 21 o
Vay so hang thuf tii cua khai trien
29.
~'^^ = 21
o n = 7
a
b^-a'^^'
la :
b-a • +
7-3
^b^-aM
a(b + a)
C?
= 35
[b-aj
b-a
^ a
J
Biet rSng tong tat ca cac he so cua khai trien nhi thufc (x^ + 1)" bang
1024. Hay t i m he so ciia so hang chufa x^^ trong khai trien do.
CHIDAN
(1 + x'r = ci +c\x' +cix' + ... + c y +... + c:y"
Cho X = 1 ta dirge : (1 + 1)" =
+ c;, + Cf, + ... +
+ ... + C;;
= 1024 = 2" = 2'*^ ^ n = 10
Do do he so cua x'^ la :
=
6!4!
= 210.
/
30.
2
Trong khai trien nhi thuTc NiutOn ' nx
14
chufa x\t rSng 5C;;-' = C'l
-I
1 ^ , X ^ 0, hay t i m so hang
x^
(Trich de TSDH khoi A -
2012)
CHI DAN
bCr
14
= Ct
n(n - l)(n - 2)
5n =
1.2.3
n(n'^ - 3n - 28) = 0
n = 7
TS. Vu The' Hifu - Nguygn VTnh Can
Thay n = 7 vao nhi thuTc Niutofn da cho t h i c6 :
1
-c*
12
.2,
I-]
X
X
2
+ ... +
V
f-1
So hang chufa x^ trong khai trien la so' hang thuf k + 1 sao cho :
7-k
12;
k
.Xy
X^
27-k
^ 2 ( 7 - k ) - k = 5=^k = 3
Vay so hang chufa x la : -C^
31.
CHI
1
7.6.5
1
^5^_35^3
1.2.3 2'
16
Tim he so cua so' hang chufa x^° trong khai trien nhi thufc NiutOn ciia
(2 + x)", biet rang
3"C° - 3""'C;, + 3"'C^ - 3"-''Cl + ... + (-1)"C,'; = 2048.
(Trich de TSDH khoi B - 2007)
vXy
D A N
Xet khai trien nhi thiifc Niutcfri :
(x - D" = c>" - c^x"-' + c'^x"-' - cf,x"-^ +... + (-1)"c;;
Cho X = 3 ta diroc :
2" = 3"c;; -3"-'c;, +3""'cf, -3"-'c^
+ ... + (-i)"c;;
= 2048
2" = 2048 = 2" => n = 11
Thay n = 11 vao khai trien (2 + x)" ta diioc :
(2 + x ) " = 2"c?, + 2^°c;jx +... + 2c;?x^°'+ c;;x"
32.
CHI
(*)
He so cua x^° trong khai trien (*) la : a^o = 2C\\ 22.
Khai trien bieu thufc P(x) = x ( l - 2 x f + x^(l + Sx)^** va viet P(x) diTdi
dang da thufc vdri luy thifa tang cua x. Hay t i m he so ciia x'' ciia da
thufc do.
D A N
Ta
CO :
x ( l - 2xy^ = x(C° - 2C^x + 2'C5'x' - 2''C^x' + 2'C5'x' - 2'C^x'^)
x ' ( l + Sx)"" = x^(C°o + 3Cj„x + 3'Cfnx' + 3'C;'nx' +
+ 3*C,'x" +3^C?„x^+... + 3 " ' C ; V ° )
=^ P(x) = C;|x + (C?o - 2C;)x' + ... + (3''C-^„ + 2''C^)x^ +... + 3^"c;°x
Vay he so ciia so hang chufa x'' la :
as =
1 f) q Q
+ 2'Ct = 2 7 . ^ ^ ^ ^ + 16.5 = 3320.
'
1.2.3
Hqc va on luyen theo CTDT mon Toan THPT
.'' 1 5
33.
CAC BAITAP lij GIAI
T i m so h a n g k h o n g chijfa x cQa k h a i t r i e n n h i thijfc N i u t o n .
if
X + —
X
.
j
DS : 924.
K h a i t r i e n va r u t gon P(x) = (x + 1)^ + (x - 2f t h a n h da thufc v d i luy
35.
K h a i t r i e n va r u t gon bieu thufc :
P(x) = ( 1 + x f + (1 + x)^ + (1 + xf + (1 + x)^ + (1 + x)^"
ta dirgfc : P(x) = aiox^° + agx® + asx** + ... + aix + ao
T i n h ag.
£>S : a« = 55.
34.
thtra giam dan ciia x. T i m he so cua cac so hang chufa x^ va x^.
DS : He so ciia x^ la : - 6 2 2 , ciia x^ la : 570.
36.
Chufng m i n h vdfi n nguyen diiOng t a c6 :
a) cL+cL+... + CL=cL+cL+... + c r .
b) c;, + 2Ci + 3Ci +... + nc;; - n2"-'.
CHI DAN
a) K h a i t r i e n P(x) = (x - 1)^" r o i cho x = 1.
b) K h a i t r i e n P(x) = ( 1 + x)". T i m P'(x) r o i t i n h P ' ( l ) .
T r o n g k h a i t r i e n n h i thufc
38.
Viet k h a i t r i e n Niutcfn, bieu thufc (3x - 1)^'', tU do chufng m i n h rSng :
37.
x- Xem thêm -