1.9 PAidU v PU (Permutations and Combinations):

 

1.9.1 PAidU(Permutations):

 

ĸ:- MAz vAqz 10 DlUgjzg. MAz bAii avz Pê 6 dg sm g zs vAqz ܥPg J smz EgP. M avPg wg J smU vUz CU CAzd ɯ zsj üĪg. sm wU g. 22 vUĪzzg bAiiavUg Pl CAzd Z J?

 

F ĸU jg PAqĻrAiĪz DQzAiP?

 

pP:-

 

FUU z n sP ASU v۪ PAqĻrAiĪz wzê.(o 1.8)

 

= 1+2+3+4 ..+n =n(n+1)/2

ASU PrĪ z Ut¹zU KUvz?

1*2=2

1*2*3 =6

1*2*3*4 = 24

 

z n sP ASU Utު Cx sPjAiį (Factorial) (n!) JAz PgAivê.

n!= 1*2*3*4.*n

1! =1

2!= 1*2=2=2*1!

3!=1*2*3=6 =3*2!

4! =1*2*3*4 = 24= 4*3!

n! = n*(n-1)*(n-2)*(n-3)*3*2*1=n*(n-1)!

n! = n*(n-1)! Or n =

 

1.9.1 Gzgu 1: A , B v C U vgUwAi zyUVg. ëU Cg F PɼVAv V PVz:-

1. MAz Eg zyUgĪAv.

2. 3 zyUgĪAv.

 

w Azsz ê Cg U j?

 

P:

1.9.1.1: Eg zyU 2 iqĪz

1. A Aiģ Az , Gz B Cx C U AiigzgƧg AAi Az . (2 zsz z. AB v AC)

2. FU B Aiģ Az , Gz A Cx C U BAi Az . (FU ţB 2 zs Qv. BA v BC)

3. FU CAiģ Az . A Cx B Aiģ CAi Az . (FU ţB 2 zs Qv. CA v CB)

 

1 ܣ

A

B

C

2 ܣ

B

C

A

C

A

B

 

F jwAiiV Ml 6 zs (=3*2) U z. (AB, AC), (BA, BC), (CA, CB)

 

1.9.1.2: 3 zyU 3 U Īz:

 

1. A Aiģ Az . FU Gz B Cx CU Mg AAi Az . (DU 2 P Qv. ABC v ACB)

2. BAiģ Az . BAi Az A Cx C . (FU 2 P Qv. BAC v BCA)

3. FU CAiģ Az . CAi Az B Cx A . (FU 2 P Qv. CAB v CBA)

 

1 ܣ

A

B

C

2 ܣ

B

C

A

C

A

B

3 ܣ

C

B

C

A

B

A

 

U Ml 6 (=3*2)zsU z.

(ABC, ACB), (BAC, BCA), (CAB, CBA)

 

1.9.1 Gz.2: ð GzguAiİ A, B, C v D UA 4 zyU PɼV Pz U z?

1. Aiiz 2 zyU

2. Aiiz 3 zyU

Uzg wAiAz Pz J zsU?

 

 

zs:

1.9.1.2.1: 2 zyU :-

 

1 ܣ

A

B

C

D

2 ܣ

B

C

D

A

C

D

A

B

D

A

B

C

 

FU U 12 zsU Qz (=4*3)

(AB, AC, AD),( BA, BC, BD),( CA, CB, CD),( DA, DB, DC)

 

1.9.1.2.2: 3 zyU :-

 

1 ܣ

A

B

C

D

2 ܣ

B

C

D

A

C

D

A

B

D

A

B

C

3 ܣ

C

D

B

D

B

C

C

D

A

D

A

C

B

D

A

D

A

B

B

C

A

C

A

B

 

FU U zgv U:

A Az EgĪ 6 (ABC, ABD, ACB ACD, ADB, ADC )

B Az EgĪ 6 (BAC, BAD, BCA, BCA, BDA, BDC)

C Az EgĪ 6 (CAB, CAD, CBA, CBD, CDA, CDB)

D Az EgĪ 6 (DAB, DAC, DBA, DBC, DCA, DCB)

U 24 (=4*3*2) zsz z.

 

n UAz r U KPPP vUzPAq erĪ zs (PAid) (permutations) nPr Az avê.

 

ï P irz jwAi gu:-

 

Gz.

Ml zyU AS(n)

w gĪ zyU(r)

Ai zsU

aĪ P

Cx gu

1.1

3

2

6

3P2

3 UAz 2 U vUzPAq irz PAid.

1.2

3

3

6

3P3

3 UAz 3 U vUzPAq irz PAid.

2.1

4

2

12

4P2

4 UAz 2 U vUzPAq irz PAid.

2.1

4

3

24

4P3

4 UAz 3 U vUzPAq irz PAid.

 

PAidAi U MAz PħzުV erĪ PĪVz.

 

1.9.2 JtPAi Ʈvv (Fundamental Principles of counting):

 

ê v ûv Eg MnU U UwgAz Jt¹. ģɬAz ûv ģU U 4 zjU v ûv ģɬAz U U 3 zjU. êz Pɮ j P e Pq Avz. Uzg, e, ɬAz ûv ģAi SAvg ģU J zjAiİ gz?

 

 

 

 

 

 

 

 

 

 

 

 

UĤ:-

ģɬAz ûv ģU 4 zjU v CAz U vĥ 3 zjU.

e Ai ɬAz g zjU AiizzgAz (A Cx B Cx C) zjAiİ Az ûv ģU gz. CAz ģU 4 zjU AiizzgAz (1,2,3 Cx 4) zjAiİ Az ģ gz.

PɼV vBSAi eAi ɬAz ûv ģAi SAvg ģU gzz zs zjU avz.

 

P. AS

ɬAz ûv ģU

ûv ģɬAz ģU

zj

1

A

1

A-1

2

2

A-2

3

3

A-3

4

4

A-4

5

B

1

B-1

6

2

B-2

7

3

B-3

8

4

B-4

9

C

1

C-1

10

2

C-2

11

3

C-3

12

4

C-4


eAi 12(=3*4) zsV g g zjU ģ vĥz. Czjw ģɬAz U U 12 zjU (=4*3) E.

 

Mn֣: MAz QAiAiģ m zsU v EAz QAiAiģ n zsU iqzzg, F Jgq QAiU MnU (m*n) zsU iqz. Ez JtPAi Ʈ vv.

 

n UAz MAz P r U jUt¹ PAidU PAqĻrAiĪ v:

 

MAz r S nUU EqVz. n U F S nUU vAPVz. n U r nUU vAĪ QAiAi n U PAid.

 

nU AS

1

2

3

(r-1)

r

vAĪ zsU

n

(n-1)

(n-2)

 

n-(r-2)

n-(r-1)

 

1.z nUAiģ n zsUAz vAz.

2. Jgq nUAiģ (n-1) zsUAz vAz.

3. g nUAiģ (n-2) zsUAz vAz.

Ezjw,

r nUAiģ vAzz zsU: {n-(r-1)} = (n-r+1)

JtPAi Ʈ vvz Pg r nUU vAzz Ml zsU:

n*(n-1)*(n-2)*(n-3)..(n-r+1).

 

Ez n UAz r U vUzPAq irz PAid. Ez nPr Az aĪg.

nPr = n*(n-1)*(n-2)*(n-3)..(n-r+1) =======(1)

 

ð Pgtz r z C n DzòzU,

nPn = n*(n-1)*(n-2)*(n-3)..(n-n+1)

= n*(n-1)*(n-2)*(n-3)..*1

nPn =n!

 

Pgt (1)g sU (n-r)*(n-r-1)*..3*2*1 jAz Ut¹ v sV.

nPr = {n*(n-1)*(n-2)*(n-3)..(n-r+1)* (n-r)*(n-r-1)*..3*2*1}/{(n-r)*(n-r-1)*..3*2*1}

= {n!= 1*2*3*n and (n-r)! = 1*2*3.*(n-r)}

nPr=

 

UĤ:

nP1=

=

= n

nP1 =n

 

 

nP(n-1)

= (nPr vz r z (n-1) Dzòz)

= n! (1!= 1)

nP(n-1)= n!= nPn

 

(n-r)! = n!/ nPr{ nPr= n!/(n-r)! }

 

ð Pgtz r=n Dzòz.

0! = n!/ nPn= n!/n! ( nPn= n! )=1

0! =1

 

Ml gA:-

 

n = n!/(n-1)!

nPn = n!

nP1 = n

nPn-1 = n! = nPn

0! =1


 

1.9.2 ĸ1 : COMPUTERJA zz J CPgU PAidAiģ PAqĻrj. EU J zU M Az gAsUvz?

 

jg:

zv z 8 CPgU. DzjAz 8!=40320 zU gaz.

ܣ

1

2

3

4

5

6

7

8

CPgU

M

C,O,P,U,T,E,R CPgUAz vAP.

 

M z ܣzlg, Gz 7 zUAz 7 ܣU vAP. (n=7).

M Az DgAsUĪ z ƺU = 7! = 5040

 

1.9.2 ĸ 2: 2,3,4,5 v 6 CAQU GAiVPAq 3 CAQAi J ASU gaz? CU J ASU?

 

jg:

PngĪ 5 CAQU: 2,3,4,5 v 6

gg

vg

r

(2,3,4,5,6) UAz

 

DzzjAz gazz 3 CAQAi ASU = 5P3 = = =60

 

ASU:

1.      r ܣz 2 EzU vg v gg ܣU 3,4,5,6 Eg zs 2 r ܣz EgĪzjAz Gz 3,4,5,6 CAPUAz (n=4) J 2 CAPUgĪ (r=2) CAPU zs ? : 4P2= = 4*3 = 12

2.      r ܣz 4 EzU vg v gg ܣU 2,3,5,6 Eg zs 4 r ܣz EgĪzjAz Gz 2,3,5,6 CAPUAz (n=4) J 2 CAPUgĪ (r=2) CAPU zs ? : 4P2= = 4*3 = 12

3.      r ܣz 6 EzU vg v gg ܣU 2,3,4,5 Eg zs 6 r ܣz EgĪzjAz Gz 2,3,4,5 CAPUAz (n=4) J 2 CAPUgĪ (r=2) CAPU zs ? : 4P2= = 4*3 = 12

Mn֣ 36(=12+12+12) ĸASU zs.

 

1.9.2 ĸ 3: 0,1,2,3 F CAQUߥAiV J 3 CAQU ASU gAiħz?

 

jg:

E n=4, r=3.

iqzz 3 CAQU AS: 4P3 = = 4!=24

Dzg, ƣ߬Az DgAsUĪ 3 CAQU ASAi, 2 CAQU ASAi DVgvz. (012 = 12,055=55 . .).

DzjAz ð GvgAz ƣ߬Az DgAsUĪ 3 CAQU ASU PAiĨP(ƣ zs Egz)

z CAQ ƣAiizU, Gz CAQU: n=3, EgĪ ܣU r=2

ƣ߬Az DgAsUĪ 3 CAQU ASU = 3P2 = 3! = 6.

0,1,2,3 F CAQU = 24-6 = 18 --> 3CAQU ASU gAiħz.

 

1.9.2 ĸ4: MAz Pn 7 g g ŸPU J zsU erz? CU U Pz 3 ŸPU MnU MAzq EgĪAv J equU iq zs?

 

jg:

 

En=7.

F 7 ŸPU er zsU = 7! = 5040.

F ŸPU A,B,C,D,E,F,G DVg. 3 ŸPU MnU EqP. C B, C, D DVg. F g ŸPU MmV H JAz PgAiĪ. DU U A,H,E,F,G JA 5 U zgv. EU er zsU: 5!=120.

E H MAz (B,C,D)ŸPU Pl. F Pn֣Ai ŸPU 3!=6 zsU erz.

DzjAz U Pz 3 ŸPU MnU EgPzg erĪ zsU = 6*120=720

 

1.9.3 PU(Combinations):

1.9.3 Gz. 1 : A , B,C U vgUwAi zyUVg. M smUsg Cg smU F jw vUAiĨQv:

1. Aiiz 2 zyU MAz avzgĪAv.

2. Aiiz 3 zyU MAz avzgĪAv.

Uzg D bAiiavUP J avU vUAiĨP?

 

 

jg P:

Gz. 1.1:

A Gzgu 1.9.1.1.1 gAiAv F PɼV 6 U zs:-

 

1 ܣ

A

B

C

2 ܣ

B

C

A

C

A

B

 

Dzg bAiiav vUAiĮ, AB = BA, BC = CB, v CA=AC

DzjAz av vUAiĮ g g UAU iv zs (AB, BC, CA).

 

Gz. 1.2: Gz. 1.9.1.1.2 g rzAv PɼV 6 PAidU zs.

 

1 ܣ

A

B

C

2 ܣ

B

C

A

C

A

B

3 ܣ

C

B

C

A

B

A

 

Dzg F ð UAU sm vUAiĮ MAz. (ABC).

 

1.9.3 Gz. 2: A B C D U 4 d vgUwAi P̼. M bAiiavUP Cg avU F PɼVAv vUAiĨQv:

 

1. Aiiz 2 zyU MAz avzgĪAv.

2. Aiiz 3 zyU MAz avzgĪAv.

 

Uzg D bAiiavUP J avU vUAiĨP?

 

jg P:

Gz.2.1: A Gzgu 1.9.1.2.1 gAiAv F PɼV 12 U zs:-

 

1 ܣ

A

B

C

D

2 ܣ

B

C

D

A

C

D

A

B

D

A

B

C

 

Dzg bAiiav vUAiĮ AB=BA, AC=CA, AD=DA, BC=CB, BD=DB v CD=DC.

g Dg UAU iv zs (AB, AC, AD, BC, BD, CD).

 

Gz.2.2: F Az Gz. 1.9.1.2.2 g rzAv PɼV 24 PAidU zs.

 

1 ܣ

A

B

C

D

2 ܣ

B

C

D

A

C

D

A

B

D

A

B

C

3 ܣ

C

D

B

D

B

C

C

D

A

D

A

C

B

D

A

D

A

B

B

C

A

C

A

B

 

Dzg bAiiav vUAiĮ,

ABC=BAC=ACB=BCA=CAB=CBA

ABD=ADB=BAD=DAB=DBA=BDA

ACD=ADC=CAD=DAC=DCA=CDA

BCD=BDC=CBD=CDB=DBC=DCB

DzjAz 24 UAUzg , bAiiav vUAiĮ g 4 UAU. (ABC, ABD, ACD, BCD)

 

n UAz r U DAi iqĪz P. (Combination)

APwPV P nCr JAz avê.

FU svA öĪ.

 

Gzgu AS

zyU Ml AS(n)

w avQgĪ zyU

Ml DAi

aĪ P

Gz. 1.1

3

2

3

3C2

Gz.1.2

3

3

1

3C3

Gz.2.1

4

2

6

4C2

Gz.2.2

4

3

4

4C3

 

FU PAidU(1.9.1) PP(1.9.3) EgĪ AAzs ð GzguU qĪ:-

 

 

Gzgu AS

zyU AS (n)

MAz j jUt¹z zyU(r)

PAid (nPr)

(1.9.1)

PU

(nCr)

(1.9.3)

nPr/nCr

=

Gz.1.1

3

2

6= 3P2

3=3C2

2=2!

Gz. 1.2

3

3

6= 3P3

1=3C3

6=3!

Gz.2.1

4

2

12= 4P2

6=4C2

2=2!

Gz.2.2

4

3

24= 4P3

4=4C3

6=3!

 

ð vBSAiAv:

nPr= nCr * r! nCr = nPrr!

 

n UAz MAz P r U jUt¹ PU PAqĻrAiĪ v:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(n UAz r U PAid) = ( n UAz r U DAi)*( r U PAid)

 

nPr = nCr* rPr

 

1.9.3 ĸ 1: nPr = 336 v nCr=56 Dzg n v rU PAqĻrj.

 

jg:

 

nPr/nCr = r!

r!= = 6=3*2*1=3!

r=3

nCr= nPrr! = {n! (n-r)} r!

= {n*(n-1)*(n-2)*(n-3)! (n-3)! }3!

56 = n*(n-1)*(n-2) 6

I.e. 56*6 =336 = n*(n-1)*(n-2) = 8*7*6

n=8

1.9.3 ĸ 2 : MAz gd CgģAiİ 8 zsU DAzz erU. CU J g g zsU erz? (ïw. P 116)

 

jg:

 

Ml erU AS (n) = 8

 

A.

erĪ P

 

1

1 erAiģ erĪ PU

8C1

2

2 erU erĪ PU

8C2

3

3 erU erĪ PU

8C3

4,5,6

-------------

 

7

7 erU erĪ PU

8C7

8

8 erU erĪ PU

8C8

 

Ml erzz PU = 8C1+ 8C2 + . . . + 8C7 + 8C8 =255 = 28-1

 

 

1.9.3 ĸ 3 : MAz ʪP âPAi UAq tU AztP iqĪ PAizz. Czg zP 5 t v 4 qUg AztPUV Azĸnzg. Eg qUg v Eg qVAig AztPAiģ Cg J zsz iq zs?

 

jg:

1.âPAiİ 4 qUjzg. Cg Eg DAi iqĪ zsU: 4C2=4*3*2!/2!*2! = 6

2.5 qVAijzg. Czg Eg DAi iqĪ zsU: 5C2=5*4*3!/3!*2! = 10

ð 6 UAUgĪ Eg qUg UA 10 UAUgĪ Eg qVAig UAAU dvUrz.

Ml iqzz AztPU = 6*10 = 60

 

1.9.3 ĸ 4 : MAz vAqz 10 DlUgjzg. MAz bAii avz Pê 6 dg sm g zs vAqz ܥPg J smz EgP. M avPg wg J smU vUz CU CAzd ɯ zsj üĪg. sm wU g. 22 vUĪzzg bAiiavUg Pl CAzd Z J?

 

jg:

ܥPg J smz EgPVgĪzjAz 5 dg vAq iqP.

n =10, r=5

10 dg 5-dg vAqUV iqzzz = 10C5

= 10!/5!*5!

= (10*9*8*7*6*5!)/(5!*5!)

= 10*9*8*7*6/120

= 9*4*7 =252 avU

bAiiavP Ml RZ= 252*22= g. 5, 544

 

1.9.3 ĸ 5 : Aiİ Az üĪ w AiP Mg CzPjzg. Utv, id eգ, ieգ, w, EAV, P, Pq, zʻP Pt. Eg Mg SåzsAig.

(a) 5 dg J wU iqz?

(b) J wU SåzsAig EgĪŢ?

 

jg:

Ml CzPg AS (n) = 8

wAiİgĪ CzPg (r) =5

iqzz Ml wU = 8C5

= 8!/(8-5)!*5!

= 8*7*6*5!/3!*5!

= 8*7*6/6 = 56

SåzsAig wAiİgPzg, Gz CzsPg = 7

CzsPg AS (n) = 7.

SåzsAig FUU wAi zgzjAz, FU 4 dg w iqP. (r) =4.

SåzsAig EgĪ wU = 7C4

= 7!/(7-4)!*4!

= 7*6*5*4!/3!*4!

= 7*6*5/6 = 35

SåzsAijz wU = Ml wU-SåzsAijgĪ wU

= 56-35 =21

 

1.9.3 ĸ6 : MAz vz ggSUvz 20 AzU. J (a) ggSU U (b) wPãU, F AzU ƮP z UĪAv JAiħz?

 

jg:

ggRܪz AzU AS: (n=20)

ggSU Pz AzU 2 (r=2),

JAiħzz ggSU = 20C2= 20!/(20-2)!*2!

= 20*19*18!/18!*2! = 20*19/2 = 190

 

wPãUU Pz AzU : r=3

gazz wPãU = 20C3= 20!/(20-3)!*3!

= 20*19*18*17!/17!*3!= 20*19*18/6 = 1140

 

 

 

 

1.9 Pv SAU

 

 

PtU

PAidU

PU

Cx ====>

U Pħzުz equ

zs U DAi

Gzgu====>

MATHS F z CPgUAz J U gaz?

20 d Q DlUgg 10 dg

vAq J zsz gaz?

S ====>

n UAz r U vUzPAq erĪ zsU.

n UAz

r U DAi iqĪz.

v====>

nPr = n(n-1)(n-2)(n-r+1)=

nCr = nPr /r!

AAzs ===>

nPr= nCr * r!