MATH Discrete Structures

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1 MATH Discete Stuctues hapte III ombiatoics Pemutatios ad ombiatios The Multiplicatio Piciple Suppose choices must be made, with m 1 ways to make choice 1, ad fo each of these ways, m 2 ways to make choice 2, ad so o, with m ways to make choice The thee ae m m L m diffeet ways to make the etie sequece of choices 1 2 Factoial Notatio Fo ay atual umbe, ( )( )! 1 2 L Also, 0! 1 Pemutatios A pemutatio of (whee 1) elemets fom a set of elemets is ay specific odeig o aagemet, without epetitio, of the elemets Each eaagemet of the elemets is a diffeet pemutatio The umbe of pemutatios of thigs take at a time (with ) is witte If P(, ) (whee P(, ) o P ) is the umbe of pemutatios of elemets take at a time, the (, ) P! ( )! Remak: P! Refeece: KH Rose, Discete Mathematics ad Its Applicatios, 5 th Editio, McGaw-Hill, 2003 Daicks ha

2 MATH Discete Stuctues s I) Thee maied couples have bought six seats i a ow fo a pefomace of a musical comedy (a) I how may ways ca they be seated? 6! 20 (b) I how may ways ca they be seated if each couple is to sit togethe with the husbad to the left of his wife? 3! 6 (c) I how may ways ca they be seated if each couple is to sit togethe? 3! 2! 2! 2! 48 (d) I how may ways ca they be seated if all the me ae to sit togethe ad all the wome ae to sit togethe? 2! 3! 3! 2 II) I how may ways ca 8 people A, B,, D, E, F, G ad H be seated i a ow if (a) thee ae o estictios o seatig aagemet; 8! (b) pesos A ad B must ot sit ext to each othe; 8!!2! III) I how may ways ca six coupos fo fee luches at diffeet estauats be distibuted amog 10 studets (a) if oe is to eceive moe tha oe coupo; P (b) if thee is o estictio o the umbe of coupos that each studet ca eceive? Remak: The umbe of -pemutatios of a set of objects with epetitio allowed is Refeece: KH Rose, Discete Mathematics ad Its Applicatios, 5 th Editio, McGaw-Hill, 2003 Daicks ha

3 MATH Discete Stuctues ombiatios A combiatio of (whee 1) elemets fom a set of elemets is a subset of elemets without egad to ode, (o ) deotes the umbe of combiatios of elemets take at a time, whee the If ( )! ( )!!, Remak: 1 ad Fo bettig o the Mak Six daw, (a) how may sigle eties ca be split fom a 8-umbe multiple ety? (b) how may sigle eties ca be split fom a 3-bake-ad--leg-umbe ety? Remak: Thee ae combiatios of elemets fom a set of elemets whe epetitio of elemets is allowed Suppose thee ae 1 ed ball, 1 blue ball ad 1 gee ball i a box Five studets ae ivited to come out oe by oe to daw a ball fom the box ad put it back How may combiatios of colos ae possible? (Note: GRBBR ad RBRGB ae egaded as the same combiatio) Refeece: KH Rose, Discete Mathematics ad Its Applicatios, 5 th Editio, McGaw-Hill, 2003 Daicks ha

4 MATH Discete Stuctues Pemutatios with Idistiguishable Objects Theoem The umbe of diffeet pemutatios of objects, whee thee ae 1 idistiguishable objects of type 1, 2 idistiguishable objects of type 2,, ad k idistiguishable objects of type k, is!!! L! 1 2 k s I) How may stigs ca be made by eodeig the lettes of the wod daicks? P! 5040 II) How may stigs ca be made by eodeig the lettes of the wod daickscha? 11! 1! 2! 1! 1! 2! 1! 1! 1! 1! 11! 2!2! III) How may stigs ca be made by eodeig the lettes of the wod daickswaihogcha? 18! ! 2! 2! 2! 2! Theoem The umbe of ways to distibute distiguishable objects ito k distiguishable boxes! so that i objects ae placed ito box i, i 1,2, K, k, equals!! L! 1 2 k s I) I a class of 20 studets, 5 of them will get Gade A, 10 of them Gade B, 3 of them Gade, ad 2 will be fail How may gade distibutios ae possible amog 20 studets? 20! 5! 10!3!2! II) How may ways ca we distibute a stadad deck of 52 playig cads ito 4 sets of 13 cads each? 52! (vey lage!!!) 13! 13! 13! 13! Refeece: KH Rose, Discete Mathematics ad Its Applicatios, 5 th Editio, McGaw-Hill, 2003 Daicks ha

5 MATH Discete Stuctues The Pigeohole Piciple Suppose that a flock of pigeos flies ito a set of pigeoholes to oost The pigeohole piciple states that if thee ae moe pigeos tha pigeoholes, the thee must be at least oe pigeohole with at least two pigeos i it Theoem [The Pigeohole Piciple] If k + 1 o moe objects ae placed ito k boxes, the thee is at least o box cotaiig two o moe of the objects Poof Suppose cotay that thee is at most 1 object i each box The total umbe of objects i the k boxes should be less tha o equal to k If evey studet i a class will eceive a gade fom A to E, ad thee ae 6 studets i the class, the thee should be at least 2 studets will eceive the same gade Theoem [The Geealized Pigeohole Piciple] If N objects ae placed ito k boxes, the thee is at least oe box cotaiig at least N k objects Poof Suppose cotay that all boxes ae cotaiig at most k 1 N N umbe of objects is at most k < + k < N k k N objects The, the total I) How may cads must be selected fom a stadad deck of 52 cads to guaatee that at least thee cads of the same suit ae chose? N 4 N 3 > 2 N > 8 ; theefoe the least N is 9 4 II) How may must be selected to guaatee that at least thee heats ae selected? 42 Refeece: KH Rose, Discete Mathematics ad Its Applicatios, 5 th Editio, McGaw-Hill, 2003 Daicks ha

Week 3-4: Permutations and Combinations

Week 3-4: Permutations and Combinations Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication