Permutations & Combinations. Dr Patrick Chan. Multiplication / Addition Principle Inclusion-Exclusion Principle Permutation / Combination

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1 Discrete Mathematic Chapter 3: C outig 3. The Basics of Coutig 3.3 Permutatios & Combiatios 3.5 Geeralized Permutatios & Combiatios 3.6 Geeratig Permutatios & Combiatios Dr Patrick Cha School of Computer Sciece ad Egieerig South Chia Uivers ity of Techology Ageda Basic Coutig Priciples Multiplicatio / Additio Priciple Iclusio-Exclusio Priciple Permutatio / Combiatio Distributig Objects ito Boxes Geeratig Permutatios & Combiatios 2

2 Why Coutig? The brute force attack is the most commo way (time cosumed but effective) i hackig How security of your password? 5 digits at most Each digit either 0-9 a-z or A-Z How may times a hacker eed to try i the worst situatio? 3 Why Coutig? Coutig problems arise throughout mathematics ad computer sciece For example the umber of experimet outcomes the umber of operatios i a algorithm (time complexity) 4

3 Basic Coutig Priciple Multiplicatio / Additio Priciple Iclusio-Exclusio Priciple Permutatio / Combiatio Basic Coutig Priciples Multiplicatio (Product) Rule If a task ca be costructed i t successive steps ad step i ca be doe i i ways where i = t the the umber of differet possible ways is x 2 x...x m 5 6

4 Basic Coutig Priciples Additio (Sum) Rule If a task ca be doe i oe of ways i oe of 2 ways... or i oe of m ways where all sets of j ways are disjoit the the umber of ways is m Basic Coutig Priciples: Multiplicatio/Additio Priciple Example I 999 a virus amed Melissa is created by David L. Smith based o a Microsoft Word macro Melissa seds a "Here is that documet you asked for do't show it to aybody else." to the top 50 people i the address book How may s are set after 4 iteratios? st iteratio: 2 d iteratio: 3 rd iteratio: 4 th iteratio: x 50 = x 50 = x 50 = (By Multiplicatio Rule) (By Additio Rule) 8

5 Basic Coutig Priciples: Multiplicatio/Additio Priciple Example 2 A programmig laguage Begier's Allpurpose Symbolic Istructio Code (BASIC) GW-BASIC (986) i MS-DOS 9 Basic Coutig Priciples: Multiplicatio/Additio Priciple Example 2 I BASIC the requiremets of a variable ame A strig of or 2 alphaumeric characters (a-z or 0-9) Begi with a letter Uppercase ad lowercase letters are ot distiguished Differet from the 5 strigs of two characters that are reserved How may differet variable ames are there i this versio of BASIC? 0

6 Basic A strig Coutig of or 2 Priciples: alphaumeric Multiplicatio/Additio characters (umber & letter) Priciple Begi with a letter Example 2 Uppercase ad lowercase letters are ot distiguished Differet from the 5 strigs of two characters that are reserved Number of variables ames cotaiig character (V ) V = 26 because a oe-character variable ame must be a letter Number of variables ames cotaiig 2 characters (V 2 ) For V 2 by the product rule there are 26 x 36 strigs of legth two that begi with a letter ad ed with a alphaumeric character However five of these are excluded V 2 = 26 x 36 5 = 93 Total umber is V + V 2 = = 957 Basic Coutig Priciples: Iclusio-Exclusio Priciple Suppose that a task ca be doe i A or i B ways But some of the set of A ways to do the task are the same as some of the B ways to do the task overcout A B A B Avoid the overcout A U B = A + B - A B 2

7 Basic Coutig Priciples: Iclusio-Exclusio Priciple Example How may bit strigs of legth 8 either start with a bit or ed with the two bits 00? Start with : 2 7 = 28 ways Ed with 00: 2 6 = 64 ways Some of these strigs are the same The bit strigs of legth eight start with a bit ad ed with the two bits = = 60 3 Basic Coutig Priciples: Iclusio-Exclusio Priciple Example 2 A computer compay receives 35 0 applicatios Suppose that 220 majored i computer sciece 47 majored i busiess 5 majored both i computer sciece ad i busiess How may of these applicats majored either i computer sciece or i busiess? Let A : the set of studets majored i computer sciece A U A 2 A 2 : the set of studets majored i busiess = A + A 2 - A A 2 = = = 34 of the applicats majored either i computer sciece or i busiess 4

8 Basic Coutig Priciples Permutatio A permutatio of a set of distict objects is a ordered arragemet of these objects st 2 d r th th (-) (-r+) =! Geeral Case The orderig of r elemets selected from distict elemets is called r-permutatio! P r = P( r) = ( )( 2)...( r+ ) = ( r)! Basic Coutig Priciples Combiatio The uordered selectio of r elemets from distict elemets is called r-combiatio It is a subset of the set with r elemets C r = C( r) = = r r!(! r)! 5 6

9 Basic Coutig Priciples Combiatio C( r) = C( - r) Algebraic Proof C( r) =! r!( r)! = ( (! r))!( r)! =C( r) 7 Basic Coutig Priciples Combiatio Combiatorial proof Usig coutig argumets to prove that both sides of the idetity cout the same objects but i differet ways Usig combiatorial proof for C( r) = C( - r) Suppose that S is a set with elemets. Every subset A of S with r elemets correspods to a subset of S with - r elemets amely A Cosequetly C( r) = C( - r) C r A = C -r A 8

10 Basic Coutig Priciples Permutatio / Combiatio Proof P 2 = 3 C 2 x 2 P 2 Number of r-permutatios of elemets P( r) =C( r) P( r r) Number of r-combiatios of elemets Number of r-permutatios of r elemets P( r) P( r r) =!/( r!/( r r)! r)! =! r!( r)! Basic Coutig Priciples: Permutatio / Combiatio Example Your class has 0 studets. How may differet ways the committee ca be set up:. A committee of four 2. A committee of four ad 0 C 4 4 C oe perso is to serve as chairperso 3. A committee of four ad two co-chairpersos 4. Two committees: 0 C 4 0 C 4 4 C 2 0 C 4 4 C 2 0 C 3 3 C Oe with four members with two co-chairs Oe with three members ad a sigle chair 9 20

11 Coutig Problems How to apply what you have lear to solve the coutig problems? Multiplicatio / Additio Priciple Iclusio-Exclusio Priciple Permutatio / Combiatio List all the possibilities 2 Coutig Problems May coutig problems ca be treated as the ways objects ca be placed ito boxes Distiguishable (labeled) Boxes Idiguishable (ulabeled) Distiguishable (labeled) Objects Idiguishable (ulabeled) 22

12 Distiguishable Objects Distiguishable Boxes How may ways are there to arrage the 5 studets ito a classroom with 5 seats? Permutatio 5! st 2 d 3 rd 4 th 5 th Distiguishable Objects Distiguishable Boxes How may ways are there to arrage the 5 studets (A B C D & E) ito a classroom with 5 seats ad the studet A ad B sit ext to each other? Two situatios: 23 AB C D E BA C D E 4! 4! 2 x 4! 24

13 Distiguishable Objects Distiguishable Boxes How may ways are there to arrage the 5 studets ito the roud table with 5 seats? A roud table meas each patter couts 5 times 5! / 5 Distiguishable Objects Distiguishable Boxes How may strigs of legth r ca be formed from the Eglish alphabet? letters Each letter ca be used repeatedly 26 r strigs of legth r 26

14 Distiguishable Objects Distiguishable Boxes How may differet strigs ca be made by reorderig the letters of the word SUCCESS SUCCESS cotais 3 Ss 2 Cs U ad E Cosiderig: 3 Ss: 7 positios are available 7 C 3 4 positios leave 2 Cs: 4 positios are available 4 C 2 2 positios leave U: 2 positios are available 2 C positio leave E: positios is available C Cosequetly the umber of differet strigs 7 C 3 x 4 C 2 x 2 C x C = 420 Distiguishable Objects Distiguishable Boxes m How may ways to give a apple a orage a baaa ad a strawberry to Mickey Mouse ad Doald Duck? 27 Mickey 0 = 4 C 0 4 C C + 4 C C C 4 Mickey = 4 C Mickey 2 = 4 C 2 Mickey 3 = 4 C 3 Mickey 4 = 4 C 4 (Doald Duck takes the rest) 28

15 Distiguishable Objects Distiguishable Boxes m How may ways to assig 3 apples ad 2 orages to Mickey Mouse ad Doald Duck? 6 x 2 = 2 Distiguishable Objects Idistiguishable Boxes How may poker hads of 5 cards ca be dealt from a stadard deck of 52 cards? 29 Combiatio = 52 r = 5 52 C 5 A = = = 2 A A

16 Distiguishable Objects Idistiguishable Boxes How may flushes are possible i stadard deck of 52 cards? Flush: A set of five cards of the same suit Four suits 4 Hearts Spades Clubs & Diamods For each suit 3 C 5 = cards are selected from 3 cards Aswer: 4 x 287 = Flush 2 A Not Flush Distiguishable Objects Idistiguishable Boxes How may ways to put 3 apples 2 orages ad baaa to 3 idistiguishable boxes ad each box cotais item?

17 Distiguishable Objects Idistiguishable Boxes m How may ways are there to put four differet employees ito three idistiguishable offices whe each office ca cotai ay umber of employees? Let A B C ad D be the four employees Oe Office 4 employees Two Offices employee 3 employees 2 employees 2 employees Three Offices 2 employees employee employee Distiguishable Objects Idistiguishable Boxes Oe Office 4 employees Two Offices employee 3 employees 2 employees 2 employees Three Offices 2 employees employee employee 4 differet ways m { {ABCD { { { { { { { { { { { { { { { { ABC { ABD { ACD { ACD {A B {A C {A D {A B {A C {A D {C D {B D {B C {D {C {B {A {C D {B D {B C {C {B {B {A {A {A { { { { { { { {D {D {C {B {C {D 33 34

18 Distiguishable Objects Idistiguishable Boxes m How may ways to put 3 apples 2 orages ad baaa to 3 idistiguishable boxes ad each box cotais 2 items? 3 Idistiguishable Objects Distiguishable Boxes How may routes are there from the lower-left corer of 4 x 3 grid to the upper-right corer if the turtle is restricted to travelig oly to right or upward? U R U U R R U Each ay route cotais 7 moves: 4 Upward (U) 3 Right (R) Aswer: 7 C

19 Idistiguishable Objects Distiguishable Boxes How may ways are there to select 5 bills from a cash box cotaiig $ bills $2 bills $5 bills $0 bills $20 bills $50 bills ad $00 bills? Assume: Order of selectig does ot matter Bills of each deomiatio are idistiguishable At least five bills of each type Idistiguishable Objects Distiguishable Boxes Select five bills from $ $2 $5 $0 $20 $50 ad $00 37 Bill Bill Bill Bill * ** * * Bill *** * * = 6 bars (lies betwee 7 boxes) 5 stars (5 bills) Total characters * * * * * C 5 =! / (5!6!) =

20 Idistiguishable Objects Distiguishable Boxes How may solutios does the equatio x + x 2 + x 3 = have where x x 2 ad x 3 are oegative itegers? Selectio of items with x items of type x 2 items of type 2 ad x 3 items of type 3 = 3 r = +3- C = (3x2) / (x2) = 78 x x 2 x 3 Idistiguishable Objects Distiguishable Boxes How may solutios does the equatio x + x 2 + x 3 = have where x x 2 ad x 3 itegers ad x x 2 2 ad x 3 3? 39 Selectio of items with x items of type x 2 items of type 2 ad x 3 items of type 3 At least item of type 2 items of type 2 ad 3 items of type = 6 have already be located Therefore =3 r = 6 = C 5 = (7x6) / (x2) = 2 x x 2 x 3 40

21 Idistiguishable Objects Idistiguishable Boxes How may ways are there to pack 3 copies of the same book ito 5 idetical boxes ad a box ca cotai book? Aswer is = = = Idistiguishable Objects Idistiguishable Boxes m How may ways are there to pack 6 copies of the same book ito 4 idetical boxes where a box ca cotai as may as six books? 4 By listig all the possibilities There are 9 ways 42

22 Geeratig Permutatios & Combiatios Sometimes permutatios or combiatios eed to be geerated but ot just couted How ca we systemically geerate all the combiatios of the elemets of a fiite set? 43 Geeratig Combiatios Recall that the bit strig correspodig to a subset For k th positio: : a k is i the subset 0 : a k is ot i the subset If all the bit strigs of legth ca be listed the by the correspodece betwee subsets ad bit strigs a list of all the subsets is obtaied 44

23 Geera tig Combia tios Next Larger Bit Strig Algorithm: Geeratig the ext bit strig (b - b b b 0 ) where the curret bit strig is ot equal to...). i = 0 2. while b i = 2. b i = i = i + 3. b i = Treat it as a biary umber ad add each time Geeratig Combiatios: Next Larger Bit Strig Example Fid out the ext combiatio usig ext larger bit strig algorithm for Next: i = 0 2. while b i = 3. b i = 2.b i = 0 2.2i = i + 45 b 3 b 2 b b 0 46

24 Geera tig Combia tios Next Larger r-combiatios Algorithm: Geeratig the ext larger r- combiatios after {a a 2 a r by give a set { {a a 2. i = r { while a i = -r+ i 2. i = i - 3. a i = a i + 4. for j = i + to r 4. a j = a i + j -i locate the last a i iea i -r + add to a i From a i+ to a r Assig ew values { 2 { 3 { 4 {2 3 {2 4 {3 4 Geera tig Combia tios Example Fid the ext larger 4-combiatio of the set { after { i = r while a i = -r + i 2. i = i - 3. a i = a i + 4. for j = i + to r 4. a j = a i + j -i a = a 2 = 2 a 3 = 5 ad a 4 = 6 The last a i such that a i -r + is a 2 (i = 2) Next larger 4-combiatio a 2 a 3 a 4 = a 2 + = a 2 + j i = a 2 + j i Hece : { = 2 + = 3 = = 4 = = 5 a 4 = 6 a 3 = 5 a 2 = 2 = =

25 Geera tig Combia tios Example 2 List all 3-combiatio for the set {a b e Assume {a b e = { 2 5 For all {a a 2 a 3. {a b c 2. {a b d 3. {a b e 4. {a c d 5. {a c e 6. {a d e 7. {b c d 8. {b c e 9. {b d e 0. {c d e. i = r 2. while a i = -r + i 2. i = i - 3. a i = a i + 4. for j = i + to r 4. a j = a i + j -i 49 Geeratig Permutatios Ay set ca be placed i oe-to-oe correspodece with the set { The permutatios of ay set of elemets ca be listed by geeratig the permutatios of the smallest positive itegers The algorithms based o the lexicographic (or dictioary) orderig is discussed 50

26 Geeratig Permutatios Algorithm: Geeratig the ext permutatio of (a a 2... a ) i Lexicographic Order by give permutatio is { 2... where (a a 2... a ) is ot equal to ( ). j = 2. while a j > a j+ 2. j := j 3. k = 4. while a j > a k 4. k = k 5. iterchage a j ad a k 6. r = 7. s = j + 8. while r > s 8. iterchage a r ad a s 8.2 r = r ad s = s + j is the largest subscript with a j < a j+ a k is the smallest iteger greater tha a j to the right of a j Sort the umber after the j th positio i icreasig order {a a 2 a 3 { 2 3 { 2 3 { 3 2 {2 3 {2 3 {3 2 {3 2 Geeratig Permutatios Example What is the ext permutatio i lexicographic order after 36254? 5 The last pair of a j ad a j+ where a j < a j+ is a 3 = 2 ad a 4 = 5 The least iteger to the right of 2 that is greater tha 2 isa s = 4 Exchage a j ad a s Hece 4 is placed i the third positio 5 2 are placed i order i the last three positios Hece the ext permutatio is

27 Geeratig Permutatios r-permutatios How ca we list all r-permutatios from a set { ? r-combiatio {a a 2 a 3 { Use ext larger r-combiatios lists all r-combiatios 2. For each r-combiatio use -permutatio to list all permutatios { 2 3 { 2 4 { 3 4 { permutatio { 2 3 { 3 2 {2 3 {2 3 {3 2 {3 2 53