8/14/2019 Chuyen e 18:

1/8

GIAI TCH TO HPChuyen e 18:

I.KHAI NIEM VE GIAI THA:1.nh ngha: Vi nNva n > 1

Tch cua n so t nhien lien tiep t 1 en nc goi la n - giai tha. Ky hieu : n!

Ta co :n! = 1.2...n

* Quy c : 0! = 1 va 1! = 12.Mot so cong thc:

* n! = (n - 1)!.n * = (k+1)(k+2)...n (n k) *n!

k!

n!(n k 1)(n k 2)..

(n k)!= + +

II.CAC QUY TAC C BAN VE PHEP EM:

1.QUY TAC CONG:V du: Co 8 quyen sach khac nhau va 6 quyen v khac nhau. Hoi co bao nhieu cach chon mot trong cacquyen o.

Quy tac cong cho trng hp hai oi tng : (Ap dung khi ta phan chia trng hp e em) Neu co m cach chon oi tng x

n cach chon oi tng yva neu cach chon x khong trung vi bat ky cach chon y naoth co (m+n) cach chon.

Tong quat:

Neu co m1 cach chon oi tng x1m2 cach chon oi tng x2.........................................mn cach chon oi tng xn

va neu cach chon oi tng xi khong trung vi cach chon oi tng xj nao (i j ; i,j=1,2,...,n)th co (m1+m2+...mn) cach chon mot trong cac oi tng a cho.

2. QUY TAC NHAN: (Ap dung khi ta phan tch viec thc hien mot phep chon ra thanh nhieu bc lientiep )

V du: An muon ru Bnh en chi nha Cng. T nha An en nha Bnh co 3 con ng. T nha Bnh nha Cng co 4 con ng i. Hoi An co bao nhieu cach i en nha Cng

141

8/14/2019 Chuyen e 18:

2/8

Quy tac nhan:

Neu mot phep chon c thc hien qua n bc lien tiep:bc 1 co m1 cach chonbc 2 co m2 cach chon-----------------------------bc n co mn cach chon

th co (m1.m2...mn) cach chon.

V du: Ngi ta co the ghi nhan cho nhng chiec ghe trong mot giang ng bang mot ch cai va mot songuyen dng khong vc qua 100. Bang cach nh vay, nhieu nhat co bao nhieu chiec ghe co thc ghi nhan khac nhau.

III.HOAN V:V du: T cac ch so 1;2;3 co the lap c bao nhieu so t nhien co 3 ch so khac nhau.

1.nh ngha :

Cho tap hp X gom n phan t (n >1).

Moi cach sap th t n phan t cua tap hp Xc goi la mot hoan v cua n phan t o

2.nh ly :Ky hien so hoan v cua n phan t la Pn , ta co cong thc:

nP n!=

V du: Mot to co 10 hoc sinh. Hoi co bao nhieu cach to nay ng thanh mot hang docIV.CHNH HP:

V du: T cac ch so 1;2;3 co the lap c bao nhieu so t nhien co 2 ch so khac nhau. 1.nh ngha:

Cho tap hp X gom n phan t . Moi bo gom k (1 phan t sap th t cua tap hp Xnk )c goi la mot chnh hp chap k cua n phan t cua X.

Hoan v Nhom co th t u mat n phan t

cua X

n phan t

142

Chnh hp Nhom co th t Gom k phan t

c lay t nphan t cua X

n phan t

8/14/2019 Chuyen e 18:

3/8

2.nh ly:Ky hieu so chnh hp chap k cua n phan t la , ta co cong thc:k

nA

k

n

n!A

(n k)!=

V du: Co bao nhieu so co 3 ch so gom toan cac ch so le khac nhau ?V.TO HP:

V du: Cho tap hp X={ .Viet tat ca cac tap con cua X gom 2 phan t}3,2,11.nh ngha:

143

Cho tap hp X gom n phan t .Moi tap con cua gom k phan t ( ) cua X0 k n

c goi la mot to hp chap k cua n phan t a cho.

2.nh ly :Ky hieu so to hp chap k cua n phan t la , ta co cong thc:k

nC

k

n

n!C

k!(n k)!=

V du 1: Mot lo hang gom 10 san pham. Hoi co bao nhieu cach chon ra 3 san pham

To hp Nhom khong co

th t Gom k phan t

c lay t nphan t cua X

n phan t

V du 2: Trong mat phang cho 7 iem, trong o khong co ba iem nao thang hang. Hoi co bao nhieu tamgiac c tao thanh .

3.Mot so cong thc ve to hp:To hp co hai tnh chat quan trong sau ay:a) vi moi k = 0,1,...,nk n

n nC C

= k 1

1

+b) vi moi k = 0,1,...,n-1k k 1 k n n n

C C C+

++ =

8/14/2019 Chuyen e 18:

4/8

VI.NH THC NIU TN:

nn 0 n 0 1 n 1 2 n 2 2 n 0 n k n k k

n n n n n

k 0

(a b) C a b C a b C a b ... C a b C a b

=

+ = + + + + =

V du 1 : Khai trien 6)2( +x

V du 2 : Chng minh rang : 0 1 2 nn n n n

C C C ... C 2+ + + + = n

LU Y QUAN TRONG:Cac bai toan ve giai tch to hp thng la nhng bai toan ve nhng hanh ong nh :lap cac so t cac so a cho ,sap xep mot so ngi hay o vat vao nhng v tr nhat nh ,lap cac nhom ngi hay o vat thoa man mot so ieu kien a cho v.v...

1. Neu nhng hanh ong nay gom nhieu giai oan th can tm so cach chon cho moigiai oan roi ap dung quy tac nhan.

2. Nhng bai toan ma ket qua thay oi neu ta thay oi v tr cua cac phan t ,th ay la nhng bai toan lien quan en hoan v va chnh hp.

3. oi vi nhng bai toan ma ket qua c gi nguyen khi ta thay oi v tr cua cac phan tth ay la nhng bai toan ve to hp.

Luyen tapBai 1: T cac ch so 1,2,3,4,5,6 co the lap c bao nhieu so co bon ch so

a) Cac ch so khong can khac nhaub) Cac ch so khac nhauc) So au va so cuoi trung nhau, khac vi 3 so gia.

Bai 2: T cac ch so 0,1,2,3,4,5 co the lap c bao nhieua) So co 3 ch so

b) So co 3 ch so khac nhauc) So chan co 3 ch so khac nhaud) So nho hn 2005, khac 0

Bai 3: Co bao nhieu cach xep 7 ngi ngoi vao mot day ban co co bay cho ngoiBai 4: Mot lp hoc co 20 nam, 10 n. Co bao nhieu cach chon 3 ngi trc lp

a) Mot cach tuy y.b) Co ung mot nc) Co t nhat mot nd) Co nhieu nhat hai n

Bai 5: Mot lp hoc co 20 nam, 10 n. Co bao nhieu cach chon mot ban can s gom 1 lp trng, 1 lp phoc tap, 1 lp pho phong traoa) Mot cach tuy yb) Lp trng la nc) Co ung mot nd) Co t nhat mot n

Bai 6: Cho n iem A1,A2,...,An thuoc ng thang a va mot iem B khong thuoc ng thang a. Noi B vA1,A2,...,An. Hoi co bao nhieu tam giac c tao thanh?

Bai 7: Tren ng tron cho n iem A1,A2,...,An.Hoi neu lay cac iem nay lam nh th:144

8/14/2019 Chuyen e 18:

5/8

a) Xac nh c bao nhieu tam giacb) Xac nh c bao nhieu t giac loi

BAI TAP REN LUYENI. CAC BAI TOAN VE PHEP EM:Bai 1:T 7 ch so 0, 1, 2, 3, 4, 5, 6 co the thanh lap c bao nhieu so chan , moi so gom 5

ch so khac nhau tng oi. KQ: 1260Bai 2: Mot to gom 8 nam va 6 n . Can lay mot nhom 5 ngi trong o co 2 n . Hoi co bao

nhieu cach chon. KQ: 840Bai 3: Cho hai ng thang song song (d1) , (d2) . Tren (d1) lay 17 iem phan biet , tren (d2)lay 20 iem phan biet . Tnh so tam giac co cac nh la 3 iem trong so 37 iem achon tren (d1) va (d2) . KQ:5950

Bai 4: T mot tap the gom 12 hoc sinh u tu , ngi ta can c mot oan i d trai he quoc tetrong o co mot trng oan , 1 pho oan va 3 oan vien . Hoi co bao nhieu cach c ?

KQ: 15840Bai 5: Xet day gom 7 ch so , moi ch so c chon t cac so 0,1,2,3,4,5,6,7,8,9 thoa man

cac ieu kien sau :- Ch so v tr so 3 la so chan

- Ch so cuoi cung khong chia het cho 5- Cac ch so v tr 4,5,6 oi mot khac nhau

Hoi co bao nhieu cach chon . KQ:2.880.000Bai 6: Ngi ta viet cac so co 6 ch so bang cac ch so 1,2,3,4,5 nh sau: Trong moi so c

viet co mot ch so xuat hien hai lan con cac ch so con lai xuat hien mot lan . Hoi cobao nhieu so nh vay. KQ:1800

Bai 7: Cho tap hp { }A 1,2,3,4,5,6,7,8=

a) Co bao nhieu tap hp con X cua tap A thoa ieu kien cha mot va khong cha 2 ?b) Co bao nhieu so t nhien chan gom 5 ch so oi mot khac nhau lay t tap A va

khong bat au bi 123? KQ: a) 64 b) 3348Bai 8: Vi 6 ch so phan biet 1, 2, 3, 4, 5, 6 co the lap c bao nhieu so co cac ch so phanbiet trong o moi so ieu phai co mat so 6. KQ: 1630

Bai 9: Co bao nhieu so t nhien gom 5 ch so khac nhau tng oi sao cho tat ca cac ch soeu khac khong va co mat ong thi cac ch so 2, 4, 5. KQ: 1800

Bai 10: Cho cac ch so 0, 1, 2, 3, 4, 5, 6, 7 co the lap c bao nhieu so gom 10 ch so cchon t 8 ch so tren , trong o ch so 6 co mat ung 3 lan , cac ch so khac co matung 1 lan . KQ: 544.320

Bai 11: Co 9 vien bi xanh , 5 vien bi o , 5 vien bi vang co kch thc oi mot khac nhau .1) Co bao nhieu cach chon ra 6 vien bi trong o co ung 2 vien bi o ? KQ:10.010

2) Co bao nhieu cach chon ra 6 vien bi trong o so bi xanh bang so bi o? KQ:4.665Bai 12: Mot hop ng 4 vien bi o , 5 vien bi trang va 6 vien bi vang . Ngi ta chon ra 4 vien bi t hop o

Hoi co bao nhieu cach chon e trong so bi lay ra khong u ca 3 mau. KQ:645Bai 13: Cho 8 ch so 0,1,2,3,4,5,6,7 .T 8 ch so so tren co the lap c bao nhieu so , moi so

gom 4 ch so oi mot khac nhau va moi so eu khong chia het cho 10. KQ: 1260Bai 14:Hoi t 10 ch so 0,1,2,3,4,5,6,7,8,9 co the lap c bao nhieu so gom 6 ch so khac

nhau sao cho trong cac ch so o co mat so 0 va so 1. KQ:42000Bai 15: Co bao nhieu so chan gom 6 ch so khac nhau tng oi mot trong o co ch so au tien la so le?

145

8/14/2019 Chuyen e 18:

6/8

8/14/2019 Chuyen e 18:

7/8

Bai 1: Giai phng trnh : )2(672 22 xxxx PAAP +=+ Bai 2: Giai phng trnh: xxCCC xxx 14966

2321 =++

Bai 3:Giai phng trnh: 41

3

1

2

4

4

1

2

=

x

xx

x

x xCCACx

04

5 22

3

1

4

1 xxx ACC Bai 4: Giai bat phng trnh:

=+

=+

8025

9052

y

x

y

x

y

x

y

x

CA

CABai 5: Giai he phng trnh:

)(53

1

11

1

1

1yx

CC

CC

y

x

y

x

y

x

y

x

=

=

++

++

+ b)

=

=+

24

1:

3

1: 2

x

y

x

y

x

y

x

y

AC

CC

Bai 6: Giai he phng trnh: a)

Bai 7: Tm cac so nguyen dng m, n thoa man: 3:5:5:: 111

1

1=++

++

m

n

m

n

m

n CCC

III. CAC BAI TOAN VE NH THC NIU-TN:

Bai 1: Tm he so cua so hang cha x43 trong khai trien21

3 2

5 1

+

xx

Bai 2: Biet tong he so cua ba so hang au tien trong khai trienn

xxx

+

15 28

3 1 bang 79. Tm so hang

khong cha x

Bai 3: Cho khai trienn

xx

+

3 2

3 3 . Biet tong he so cua ba so hang au tien trong khai trien tren

bang 631. Tm he so cua so hang co cha x5.

147

Bai 4: Tm gia tr cua x sao cho trong khai trien cua ( n la so nguyen dng ) co so hangn

x

x 1

12

2

+

th 3 va th 5 co tong bang 135, con cac he so cua ba so hang cuoi cua khai trien o co tong bang22

Bai 5: Tm so hang khong cha x trong khai trien :9

2

121)(

+=x

xxP

Bai 6: Chng minh rang: viknk

n

k

n

k

n

k

n CCCCC 3321

33 + =+++ nk3

Bai 7: Chng minh rang :2

)1(.........3.2

112

3

1

2

1 +=++++++

nn

C

Cn

C

Ck

C

C

C

CC

n

n

n

n

k

n

k

n

n

n

n

n

n

Bai 8: Chng minh rang : nnnn

n

n

n

n

n

nCCCC 97....7.2.7.22 2221110 =++++

Bai 9: Chng minh rang :1112231201 5.3....3.2)2(.3.2)1(2 =++++

nnn

nn

nn

nn

n nCCnCnCn

Bai 10: Chng minh rang:1

13

1

2...

3

2

2

22

112

31

20

+

=

+++++

++

nC

nCCC

nn

n

n

nnn

Bai 11: Tnh tong : 20052005

2

2005

1

2005

0

2005...32 CCCCS ++++=

Bai 12: Tm he so cua so hang cha x26 trong khai trien nh thc Niutn cuan

xx

+ 7

4

1, biet rang

12...20

12

2

12

1

12=+++ +++

n

nnn CCC

8/14/2019 Chuyen e 18:

8/8

Bai 13: Tnh tong1

1

1

3

2

1

2

1

1

1

0).1(

....3.2.1

+

+++++=

n

n

nnnn

A

Cn

A

C

A

C

A

CS , biet rang 211210 =++ nnn CCC

Bai 14: Khai trien bieu thc ta c a thc co dang . Tm he so cua xnx)21( nnxaxaxaa ++++ ...2

210

5

biet 71210 =++ aaa

Bai 15: Tm he so cua trong khai trien cua829yx ( )153 xyx Bai 16: Tm n sao cho :N 256... 2

24

2

24

1

24

0

24=++++ ++++

n

nnnn CCCC

Bai 17: Tm so t nhien n sao cho : nnnCCC 654

111 =

Bai 18: Chng minh rang nnnnn

n

nn

n

n

n CCCCCC ...)1(...3310113 ++=++

Bai 19: Cho 103202

)1

()1

(x

xx

xA += . Sau khi khai trien va rut gon th bieu thc A se gom bao nhieu so

hang?Bai 20: Tm so t nhien n thoa man ang thc sau:

)12(233...3...3161522

2

2222

2

22

2

22

2

0

2+=++++++ nnn

nn

n

kk

nnn CCCCC

-----------------Het-------------------

148