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Combinatorics: Rules for Working with Groups

Learn the basic definitions and rules for permutations, variations, combinations, and more in combinatorics. Discover how to solve different types of problems using these principles.

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Combinatorics: Rules for Working with Groups

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  1. Combinatorics. Therulesforworkingwithgroups: • elementsselection • subgroupsorganization Basic definitions.

  2. Example. How many differentfive-digit natural numberscanbewrittenusingthenumbers 1,2,3,4,5, if: • digits in thenumberisusedonlyonce? • How many oftypednumberswillbeginwith a number 5? • How many of the numbers written will be even? Solution. a) P(5) = 5! = 5.4.3.2.1 = 120 b) P(4) = 4! = 4.3.2.1 =24 c) ended2: P(4) = 4! =24 ended4: P(4) = 4! = 24 Finally S = 2.4! = 2.24 = 48 Factorials and binomialcoefficients. , 0 ≤k ≤ n, k  N, n N

  3. Rulesforcalculatingthebinomialcoefficients. Example. Whichkisthesolutionoftheequation? ? k + 1  2, k  1 k  2 k  2 k = 2, as k N

  4. Permutation. Permutationwithoutrepetitionfromnelementsiseach arrangement ofn-element set M.Numberofallpermutationsofthe set withnelements: Proofofthepreviousformula (mathematicalinduction). Example. On thewayfrom Kladno to Brno via Prague wecan use: from Kladno to Prague - bus, train, own car: from Prague to Brno - bus, train, private automobile, theairplane. What varioustransport optionsfrom Kladno to Brno wehave? Example. How many anagramscancreateletters m, a, t, h, e, m, a, t, i, c, s? P(11) = 11! = 39916800 Thenumberofpermutationswithrepetitionfromnelements, whereelements are repeatedsuccessivelyk1, k2, … , kn – timesis

  5. , Example. How many wayscanbedistributedamong 30 studentstwo free tickets to the concert, fivetickets to theswimming stadium and ten tickets to thegym, whereeach student canreceive a maximum ofoneticket? We have few tickets, some students will notgetanything ⇒ theygetblankpapers⇒ 2 tickets to the concert, 5 petalsintothe pool, 10 tickets to thegym and 13 empty, total) = 4.89109E+13 possibilities. 30!/(2!5!10!13!

  6. Variations. Variationoftheclasskfromnelementswithoutreplicationiseach orderedsubsetofthe set M withk elements(M iswithnelements). Thenumberofvariationsoftheclasskfromnelementswithoutreplitationis: , 0 ≤ k ≤ n, k N, n  N. Example. M = {1,2,3}. Definethenumberoforderedpairswithoutreplicationsfrom these elements. V2(3): (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2), wehave 6 variationsoftheclass 2 fromsixelements. Example. Wehavedigits1, 2, 3, 4, 5, 6. How many 3-digit orderednumberscanwecreate? Thedigits are not repeated.

  7. VariationoftheclasskfromnelementswithrepetitionsiseachorderedsubsetVariationoftheclasskfromnelementswithrepetitionsiseachorderedsubset of M withk elements, whereeach element canberepeatedktimes. Thenumberofvariationsoftheclasskfromnelementsis: , k N, n N. Example. Proofthepreviousrelation. Example. Wehavedigits1, 2, 3, 4, 5, 6. How many 3-digit numberswecancreate? Thedigits canberepeated and itdepends on thedigitsorder. Example. Anadulthuman has 32 teeth. Howlargemustbe a groupofpeopleto in itoccurred atleast twopeoplewiththesame set ofteeth.

  8. Example. Morse codeconsistsofsymbolsthatcontain· and -. Ifweconsider "thewords" for a maximum of 6 symbols, how many differentwordsavailableto us. Wehave 3-digit numbersconsistingofdigits0, 1, 2, . . . , 9. How many different Numberswecanwrite, a) thenumbercan start withzero b) thenumbercannotstart withzero. Combinations. Combinationoftheclass k from n elementswithoutreplicationsiseachunorderedsubsetwithk elementsofthe set withnelements. Theelements are not repeated. Thenumberof these combinations are: , 0 ≤k ≤ n, k  N, n N

  9. Example. Howmany 4-tonal chordscanbeplayedfrom 7 tones? Combinationoftheclasskfromnelementswithrepetitioniseachsubsetwith kelementsofthe set M withnelementsthatisunordered and each element ofit canberepeatedk-times. Thenumberof these combinationsis: , k  N, n N Example. Theshop has threecolorsofyarn. Theblobshaveweightof 50 grams. I need 500 g ofyarn. How many wayscan I buy 500 grams? Ten timeschooseoneblobfrom a set thatconsistsof 3 colorsofyarn. Thereforen = 3, k = 10.

  10. Example. Weneed to buy12 bottlesofmineralwater. Theshopsells 4 typesofwatter. How many optionswehave? n= 4, k = 12 withrepetition. Příklad. How many wayscandistribute 20 tickets to thepremiereofthe film among 10 people? 1 person canobtain0, …, 20 tickets. Thereforewemustcomputecombinationwith repetition. n = 10, k = 20,

  11. Binomialtheorem. , a R, b  R, n  N The k-thmemberoftheseriesis Example. Whichmemberoftheseriesdoes not obtainx? , x 0 Ifthismemberdoes not includex, pak x15-3k= 1, or15 – 3k = 0. Thereforek = 5.

  12. Pascal’s triangle. 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

  13. Example. Compute, wherenis natural number or n is equal to 0. In binomial theorem we havea = b = 1. Therefore= 2n. Consequence. definesthenumberofallsubsetswithkmembersofn-member set (k = 0 isempty set). The sum definesthenumberofallsubsetsof set withn members. The set withn membershavetherefore2nsubsets.

  14. EXamples. 1. Oneman has 5 coats, 4 jackets and 6 trousers. How many differentwayscanwear? 2. Howmany differentthrowsthreedicecanbe done? 3. How many 6-digit numberscanwecreatefromthedigits 1,2,3,4,5,6? Eachdigitoccursonlyonce. 4. What natural numberissolutionoftheequation 5. Whichofcombinationnumbercanberepresented as 6. Simplify 7. Ifweenlargethenumberofelements by two, thenumberofpermutationswithoutrepetitionisincreased twelvetimes. Whatisthefirstnumberofelements?

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