ChaPTer Combiatorics ChaPTer CoTeTS a The additio ad multiplicatio priciples b Permutatios C Factorials D Permutatios usig P r e Permutatios ivolvig restrictios F Arragemets i a circle G Combiatios usig C r h Applicatios to probability DiGiTal DoC doc-983 0 Quick Questios a The additio ad multiplicatio priciples itroductio Cosider how may ways two differet letters ca be listed from the letters C, A ad T if the order i which the letters are to be listed is ot take ito accout. We would write CA, CT ad AT. If the order of listig the two letters is take ito cosideratio, there will be 6 differet possibilities, amely, CA, AC, CT, TC, AT ad TA. I this chapter we itroduce some methods that will eable us to effectively determie the umber of possible ways objects ca be ordered accordig to give coditios, without ecessarily havig to list them. Combiatorial theory is widely applied i areas such as computer system desig, geetics, statistics ad probability, where arragemets are of particular importace. The additio priciple To reach the top of the hill, Jack ad Jill ca use public trasport (tram or bus) or private trasport (hire car, taxi or motorbike). I how may ways ca Jack ad Jill go up the hill if oly oe form of trasport is to be used for the etire trip? Sice the modes of trasport are mutually exclusive (that is, two forms of trasport caot be used at the same time), there are + 3 differet ways of travel. This straightforward method of summig is the additio priciple, which ca be stated as: If two operatios ca be performed i A or B ways respectively, the both operatios ca be performed together i A + B ways. WorkeD example A particular mathematics problem ca be solved i ways usig aalytical methods, i 4 ways usig approximatio techiques ad i 3 ways by trial ad error strategies. I how may ways ca the problem be solved? ChaPTer Combiatorics 9
List the give iformatio. Aalytical Approximatio 4 Trial ad error 3 Use the additio priciple as the three methods of solvig the problem are mutually exclusive. The total umber of ways is + 4 + 3 9. WorkeD example A stack of playig cards cotais four jacks, four quees ad four kigs. Gary has two jacks, a quee ad a kig i his had. But, to complete his had, Gary requires three jacks, two quees or two kigs. I how may ways ca he complete his had? List the cards remaiig i the stack. Two jacks, three quees ad three kigs remai i the stack. Gary requires a jack or a quee or a kig to complete his had. Use the additio priciple to calculate how may ways he could complete his had, give the cards that are remaiig. + 3 + 3 8 There are eight ways for Gary to complete his had. The multiplicatio priciple Suppose 4 colours are available to spray-pait 3 differet cars. Let O be the first operatio selectig a car ad let O be the secod operatio pickig a pait colour. Also let C, C, C 3 ad P, P, P 3, P 4 deote, respectively, the cars ad the available colours. The differet ways i which the job ca be udertake are {C P, C P, C P 3, C P 4, C P, C P, C P 3, C P 4, C 3 P, C 3 P, C 3 P 3, C 3 P 4 }. Sice the choice of a particular car is idepedet of the colour selected, the total umber of possibilities ca be obtaied by multiplyig together the umber of choices available from the two operatios. That is, there are 3 4 differet ways possible. The tree diagram at right shows the differet outcomes. The product of the umber of outcomes from each operatio provides the total umber of possible outcomes of the operatios performed sequetially. This method is the basis of the multiplicatio priciple, which states: if two operatios ca be performed i A ad B ways respectively, the both operatios ca be performed i successio i A B ways. Operatio O Operatio O C C C 3 P P P 3 P 4 P P P 3 P 4 P P P 3 P 4 WorkeD example 3 Juaita has to choose a outfit to wear to a party. She has 6 skirts, jumpers ad 8 shirts to choose from. If ay combiatio of these items will be acceptable attire, i how may styles of dress ca Juaita atted the party? Choose a skirt, a jumper ad a shirt. There are 6 skirts, jumpers ad 8 shirts. Use the multiplicatio priciple. The total umber of ways is 6 8 40. 30 Maths Quest Mathematical Methods CAS
We ca also represet the sequece of operatios of the above example usig boxed umbers as follows. Skirts Jumpers Shirts 6 8 40 Each box cotais the umber of possible outcomes associated with the particular operatio. WorkeD example 4 From a cafeteria 4-course luch meu, I ca choose 3 varieties of soup, types of seafood, 4 kids of side dish ad types of salad. a How may differet dishes are offered? b How may differet luches ca be ordered if oe dish from each course is selected? c How may differet types of dish are possible if soup ad seafood must be icluded with each order? TUTorial eles-44 Worked example 4 a Write the umber of dishes for each course ad use the additio rule. b There are 3 soups, seafoods, 4 side dishes ad salads. Use the multiplicatio rule (as you must sequetially order a soup, seafood, a side dish ad a salad). c Cosider the possible orders cotaiig soup ad seafood: soup ad seafood oly a The 4-course meu offers 3 + + 4 + 4 differet dishes. b Soup Seafood Side dish Salad 3 4 0 Number of differet luches is 0 c Soup Seafood 3 3 soup ad seafood ad a side dish oly Soup Seafood Side dish 3 4 3 4 60 soup ad seafood ad a salad oly Soup Seafood Salad 3 3 30 soup ad seafood ad a side dish ad a salad. Soup Seafood Side dish Salad 3 4 3 4 0 Calculate the umber of dishes possible for each order. 3 Use the additio rule to fid the total (as you ca oly order the first, secod, third or fourth combiatio). Number of differet types of dish possible + 60 + 30 + 0 exercise a The additio ad multiplicatio priciples We Juicy Chickes offers 0 varieties of roast chicke dish, 6 types of fried chicke ad types of chicke pie. How may differet chicke meals are sold by Juicy Chickes? Freda Frog eats varieties of fly o the first day, varieties o the secod day, 9 varieties o the third day ad 4 varieties o the fourth day. Assumig Freda will ever cosume of the same variety of fly ad that her daily eatig habits follow this defiite patter, fid how may flies she will eat altogether i a week. ChaPTer Combiatorics 3
3 A suburba mall cosists of five shops: Tee Fashio, Harry s Takeaway, Video & Games Arcade, Toy Palace ad Byte Computers. O a busy weeked, people wet ito Tee Fashio, 7 bought food from Harry s Takeaway ad 9 people etered the Toy Palace. Each perso visited oly oe store. How may customers did Tee Fashio, Harry s Takeaway ad Toy Palace have altogether? 4 mc Two pieces of timber ca be held together usig adhesives, fasteers or clamps. The adhesives are PVA glue, Liquid Nails ad Bodcrete. Fasteers that ca be used are ails, screws, rivets ad bolts. There are two differet types of clamp available: SureGrip ad Hold-tite. If oly oe adhesive fasteer or clamp is required, the umber of ways two pieces of timber ca be joied is: a b 4 C 3 D 4 e 9 We From a pack of playig cards, the quee of spades, kig of clubs ad quee of clubs are draw. I how may ways ca aother card from the deck be draw so that there will be three quees or two kigs? 6 mc There are 4 ovels, 7 comic books ad biographies o a bookshelf. Zoe selects ad reads ovels, 3 comics ad a biography from the shelf. However, her readig assigmet requires that she read 3 ovels, 4 comics or biographies. I how may ways ca she select books from the shelf to meet the miimum requiremets of the assigmet? a 6 b 7 C D 3 e 4 7 We3 Jack ad Diae are preparig for their weddig. They will decide o oe of 3 churches, oe of available receptio cetres ad oe of 0 holiday destiatios. How may combiatios of church, receptio cetre ad holiday are possible? 8 Alaa lives i Melboure ad iteds to go to Sydey via Caberra. She will get to Caberra by bus, cotiue o to Sydey by hire car ad retur home by air. If 4 bus lies are available for the outward jourey to Caberra, 6 car retal agecies ca be used to get from Caberra to Sydey ad 3 airlies are available for the retur trip, determie how may differet ways Alaa ca make the trip to Sydey ad back. 9 At Burpies restaurat the special meal cosists of a choice of oe of etrées, oe of 3 mai meat dishes ad oe of 4 kids of dessert. For a surprise feast at Belchies restaurat you ca have oe of differet etrées, select from 4 mai meals ad decide which oe of 3 kids of dessert to order. a How may differet combiatios of dishes are possible i a special meal cosistig of a etrée, a meat dish ad a dessert? b Fid how may differet combiatios of dishes are available to a customer who visits both places ad orders a special meal ad a surprise feast. (Assume that the customer must have a etrée, mai meal ad dessert for the surprise feast.) 0 mc O a detist s waitig room table are 3 piles of readig matter. The first pile cosists of 6 differet copies of News, the secod pile has differet issues of Geographic ad the third pile comprises 0 differet Woma s World magazies. A patiet radomly chooses oe item of readig from each pile. The umber of ways of choosig the 3 items is: a b 30 C 6 D 9 e 300 mc A Whoppa pizza base is made usig oe of types of cheese ad oe of toppigs. Up to 3 additioal toppigs are available at extra cost. The umber of differet Whoppa pizzas that ca be made cotaiig at least oe additioal toppig is: a b 6 C 4 D 8 e 0 We4 A school offers Eglish, Maths, Laguage ad Sciece as part of the curriculum. Jaice must do at least oe of these subjects. a List the differet ways Jaice ca select at least oe subject. b I how may ways ca this be doe? 3 To get to school, Eri ca walk, take the trai or catch the bus. After school she ca either walk or catch the bus to get back home. a List the differet combiatios of travel for Eri to get to school ad to retur to her home. b Show the differet travel methods as a tree diagram. 3 Maths Quest Mathematical Methods CAS
4 A hot dog cosists of a sausage i a bu with sauce. Oio, tomato, pieapple ad cheese are available as extras. How may differet types of hot dog ca be made? Durig a special morig recess, teachers had a choice of tea, orage juice, coffee, pies, cheese, salami, biscuits ad cake. However, a teacher could sample oly two kids of food ad oe drik. How may differet combiatios of two kids of food ad drik were possible? b Permutatios A permutatio is the arragemet of objects i a specific order. Awardig a first ad secod prize to two people radomly selected from a studio game-show audiece or determiig the umber of ways a group of people ca queue for tickets are examples where the order of objects eeds to be take ito accout. Cosider ow how may ways two letters ca be take from the letters B, L, U ad E ad the arraged. If the order of the letters is take ito accout ad repetitio of letters (that is, BB, LL etc.) is ot allowed, we have the possible arragemets show below: BL, LB, BU, UB, BE, EB, LU, UL, LE, EL, UE, EU We ca obtai the same result usig the multiplicatio priciple. There are 4 choices for the first letter because there are 4 letters available. Oce the first letter has bee chose there are 3 letters to choose from for the secod letter. First letter Secod letter 4 3 Notice that the multiplicatio priciple takes ito accout the order of the outcomes. That is, BL is ot cosidered to be the same as LB, BU is ot the same as UB ad so o. WorkeD example Josie picks up a Mathematics textbook, a Eglish ovel ad a Biology otebook ad places them o a shelf. Determie the umber of ways the books ca be arraged. List the ways they ca be arraged. There are three positios to be filled o the shelf. 3 There are three choices of book for the first positio o the shelf. This leaves two choices for the secod positio ad oe choice for the third positio. 3 Use the multiplicatio priciple. 3 6 arragemets 4 Let M be the Mathematics textbook, E the Eglish ovel ad B the Biology otebook. The arragemets are MEB, MBE, BME, BEM, EMB, EBM. WorkeD example 6 I how may ways ca at least two letters be chose from the word STAR if the order of the letters is take ito accout ad repetitio of letters is ot allowed? There are 3 mutually exclusive evets: choose letters from 4 letters, 3 letters from 4 letters, or 4 letters from 4 letters. For the first evet there are 4 choices for the first letter ad 3 choices for the secod letter, because repetitios are ot allowed. letters 4 3 ways ChaPTer Combiatorics 33
3 For the secod evet there are 4 choices for the first letter, 3 choices for the secod letter ad choices for the third letter. 4 For the third evet there are 4 choices for the first letter, 3 choices for the secod letter, choices for the third letter ad choice for the fourth letter. Use the additio rule to fid the total umber of possibilities. 3 letters 4 3 4 ways 4 letters 4 3 4 ways Number of ways + 4 + 4 60 WorkeD example 7 How may ways are there for differet prizes or 3 differet prizes to be awarded to a group of people if: a a perso may receive more tha oe award? b a perso may ot receive more tha oe award? TUTorial eles-4 Worked example 7 a Use the multiplicatio priciple to fid the umber of ways for prizes to be awarded. Ay oe of the people ca receive the first prize ad ay oe of the people ca receive the secod prize because the same perso may receive more tha oe prize. Use the multiplicatio priciple to fid the umber of ways for 3 prizes to be awarded ad remember that the same perso may receive more tha oe prize. 3 Use the additio rule to obtai the total umber of ways to distribute or 3 prizes. b Use the same method as above, but repetitio is ot allowed, so the umber of people to choose from is reduced each time. a prizes 3 prizes st d st d 3rd Number of ways to distribute or 3 prizes + 0 b prizes 4 0 3 prizes 4 3 60 Number of ways to distribute or 3 prizes 0 + 60 80 exercise b Permutatios We A chef restocks her collectio of spices by placig jars of pepper, utmeg, giger ad mit o the shelf. I how may differet ways ca the 4 jars be placed i a straight lie? I how may ways ca 6 studets lie up at the school catee? 3 If there are 8 competitors i a race, i how may ways ca the first three places be awarded? 4 To cacel a electroic alarm, a -digit code umber must be etered ito the code box. Assumig that digits may be repeated, how may codes are possible? Five items of mail are to be placed i letterboxes. I how may ways ca this be doe if o letterbox is to cotai more tha oe item? 34 Maths Quest Mathematical Methods CAS
6 A history quiz cosists of matchig 8 coutries with their capital cities. I how may ways ca a cotestat aswer the quiz by radomly matchig each coutry with a capital city? 7 How may ordered subsets cosistig of two letters ca be chose from the word SUPERBLY if: a a letter may be used more tha oce i each subset? b choosig the same letter more tha oce is ot permitted? 8 We6 I how may ways ca at least two letters be chose from the word MATHS if the order of the letters is take ito accout ad repetitio of letters is ot allowed? 9 We7 Calculate the umber of ways that 3 or 4 prizes ca be awarded to a group of people if: a a member of the group is allowed to receive more tha oe prize b a member of the group caot receive more tha oe prize. 0 Decide i how may ways or 3 letters ca be selected from the vowels of the alphabet if a vowel ca appear oly oce i each selectio. Determie how may umbers greater tha 0 ca be made usig all of the digits 4, 7,, 6 ad if each digit caot be used more tha oce. How may umbers greater tha 00 ad less tha 0 000 may be formed usig the digits, 3, 4 ad if each digit may be used more tha oce? 3 mc The umber of 3-digit ad 4-digit umbers greater tha 00 that ca be formed usig the digits, 6,, ad 3, if each digit ca be used more tha oce i each selectio, is: a 600 b 00 C 67 D 7 e 40 4 mc Juliaa has saved her pocket moey to buy up to 3 fashio magazies. If there are differet magazies to choose from, the umber of ways she ca buy, or 3 magazies is: a 90 b 80 C D 70 e 8 mc The total umber of -digit, 3-digit ad 4-digit odd umbers that ca be formed usig the digits,, 3, 4 ad 9, whe a digit may be used more tha oce i a group, is: a 78 b 8 C D 68 e 7 6 How may 7-letter arragemets are possible usig the letters of the word DECAGON if the letters A, E ad O must occupy the third, fifth ad sixth positios respectively ad the letters remaiig may be used more tha oce? 7 A school fudraisig competitio that costs cets per etry ivolves tryig to correctly match 9 teachers with their baby photographs. Wasim wats to be certai to wi the $000 first prize by tryig all possible combiatios. Decide how much moey Wasim will wi or lose if he is to be the prize wier. 8 A versio of the party game musical chairs has the players march aroud a lie of chairs ad scramble to sit o them whe the music uexpectedly stops. At each stage the umber of players is oe more tha the umber of chairs. The player who remais stadig whe the music stops is out of the game ad oe chair is the removed. The player remaiig sittig after all the other players have bee elimiated is the wier. a If players are takig part, how may differet arragemets of seatig are possible durig the: i first roud? ii fourth roud? b The rules are chaged so that chairs are removed each time. If there are 9 players ad 7 chairs at the start of the game, how may seatig arragemets are possible for all the rouds? 9 A school is usig idetificatio cards (ID cards) that cosist of 3 letters selected from A to E iclusive followed by 3 digits chose from 0 to 9 iclusive. a How may differet ID cards ca be issued to studets if a digit may be used more tha oce but all 3 letters of each ID are differet? b New ID cards are issued to all studets each year ad the old cards discarded. However, the old ID umbers are ot used agai. If, o average, the school s populatio icreases by 0% each year ad was 000 durig the year whe the ID cards were first used, how may years will elapse before cards with umbers already used will have to be issued? ChaPTer Combiatorics 3
C Factorials Expressios obtaied by usig the multiplicatio priciple frequetly cotai the product of cosecutive whole umbers. It is coveiet to adopt a shorthad way of represetig such expressios to assist with calculatios ad to effectively display the properties associated with permutatios ad other types of order of objects. Particularly useful is to defie! to mea the product of cosecutive positive itegers startig from dow to. That is:! ( ) ( ) ( 3)... 3 The symbol! is read as factorial. For example, 4! 4 3 4, 3! 3 6,!. Alteratively, 4! 4 3! 4, 3! 3! 6,!!.! Thus from the defiitio we have! ( )! or If we substitute we have:! ( )! or ( )! (sice! ) (0)! This expressio is true if 0! is take to be equivalet to. So we defie 0!. WorkeD example 8 a Express 7! as a umeral. b Simplify! + (3 )! a Use the defiitio of! with 7. a 7! 7 6 4 3 Multiply the umbers i the expressio obtaied. 040 b Write the expressio ad simplify (3 )! b! + (3 )!! + 6! Calculate! ad 6! 0 + 70 3 Evaluate. 960 WorkeD example 9 Simplify 8! 3! Divide the aswer for 8! by the aswer for 3! usig a calculator. 8! 8 7 6 4 3! 3! 3! 8 7 6 4 670 WorkeD example 0 a Evaluate 8! b Simplify 00! 98! a Use the factorial (!) feature of a CAS calculator to evaluate the expressio. a 8! Record the result. 40 30 3 State the aswer. 8! 40 30 36 Maths Quest Mathematical Methods CAS
b 00! ad 98! are too large to write i fully expaded form. Express 00! with 98! as a factor. 00! 00 99 (98... 3 ) 00 99 98! b 00! 98! 00 99 98! 98! 3 Cacel 98! i the expressio. 00 99 9900 Notice that:! ( )! ( ) ( )! ( ) ( ) ( 3)! etc. exercise C Factorials We8a Evaluate: a 4! b 9! c! d 3! +! e! 4! f 7! 6!! g 6! (! +! + 3! + 4! +!) We8b Evaluate: a 4 3! 4! b (4 +!) 3! +! c 6! 6! d 7 7! (8! 7!) e 8! + 3!! f 7 9! + 3 3! 9 8! g (! 4!) + (8! 7!) h! + 6!! 3 4!! + 3! + 4! 3 mc The value of is:! + 3! a b 4 C 7 D 4 e 8 4 mc The value of 4(4! 3) +!(! 4!) is: a 0 b 3 C 84 D 76 e 90 We9 Simplify: 4! a! 6 We0 Simplify: 0! a 00! 0! d 47! 7 Simplify:! a 4! 000! d 998! (+ 37)! 8 mc (00 3)! b! 4! is equal to: b e c 000! 998! 396! 393! 7! 3! 8! b 6! 63! e (936 87)! d 6! 3! c f c 400! 4499! 000! 4999!! 3! e 3!! +! 0! a 30 b 87 C 840 D 030 e 3 9 Evaluate each expressio. a 7! 4! 9! 3! 3! 6! + b + c 4! 3! 7! 3! 0! 3! d 80! 0! 77! 64! 6! 6! 8!! 6! 78!! 77! 0 Simplify each expressio. a!! 6! 7!! 8! + 4!! 8!4! 8!0! b c d e 3!! + 3!!! + 3! 3! 7!! 9!9! DiGiTal DoC doc-984 WorkSHEET. ChaPTer Combiatorics 37
D Permutatios usig P r A permutatio is a arragemet of objects i which order is importat. Cosider the letters A, B ad C. There are 6 possible arragemets or permutatios of these three letters: ABC ACB BAC BCA CAB CBA We could have determied that there are 6 possible arragemets without listig all of them usig the multiplicatio priciple, where each box below represets a positio (first letter, secod letter, third letter): st letter d letter 3rd letter 3 6 ways Note that we had 3 possibilities for the first letter but, havig placed it, we were left with possibilities for the secod letter ad, i tur, just possibility for the third. But what if we had 0 differet letters ad wished to select a arragemet of 3 letters? Agai, we could cout the umber of arragemets as follows: st letter d letter 3rd letter 0 9 8 70 ways We ca express the above calculatio usig factorials as follows: 0 9 8 7 6 4 3 0 9 8 7 6 4 3 0! 7! 0! (0 3)! Followig o from this, we ca geeralise a formula for the umber of arragemets (permutatios) of objects, takig r at a time, which we deote by P r :! P r, where ad r are atural (coutig) umbers, ad r. ( r)! Aother way of thikig of P r is as! expaded to r places. P r ( )( )...( r + ) r valuesmultiplied together I the precedig example, which ivolved arragig 0 ( 0) objects (letters) takig 3 (r 3) at a time, we ca verify that ( r + ) (0 3 + ) 8, which was the last value i the chai of multiplied umbers. Special cases. If r 0, the P r P 0!! This implies that there is oe way of selectig zero objects from objects.. If r, the P r P! 0!! There are! ways of arragig objects take from objects. 38 Maths Quest Mathematical Methods CAS
WorkeD example Calculate 7 P 3. Evaluate usig the defiitio P r. 7 7! P 3 (7 3)! 7! 7 6 4! or 4! 4! 040 7 6 4 0 0 WorkeD example Oly the 3 fastest cars i a car rally of 0 competitors will compete i the world champioships. How may differet arragemets of the 3 fastest rally cars are possible? We wat the umber of permutatios whe 3 objects are selected from 0 objects. Use P r with 0 ad r 3. Number of arragemets 0 P 3 0 9 8 70 3 Alteratively, use the permutatios feature of the CAS calculator with 0 ad r 3. 4 Record the result. 70 P r (0, 3) WorkeD example 3 How may umbers greater tha 000 ca be formed usig the digits 4,, 6, 7, 8 ad 9 if a digit caot be used more tha oce? Each 4-digit, -digit or 6-digit umber formed will be greater tha 000. Fid the umber of ways the required umber of digits ca be chose from the 6 digits give. 6 P 4 + 6 P + 6 P 6 3 Add the 3 aswers. ( or situatio). 360 + 70 + 70 800 There are 800 umbers greater tha 000 that ca be formed. WorkeD example 4 A captai ad vice-captai are to be chose from a group cosistig of 0 cricket players. From the remaiig 8 players, 3 will be selected to be the wicket keeper, spi bowler ad fast bowler. Calculate how may differet ways the positios ca be allocated. TUTorial eles-46 Worked example 4 Fid the umber of ways i which objects (captai/vice-captai) ca be chose from 0 objects (0 cricket players). 0 P ChaPTer Combiatorics 39
Fid the umber of ways i which 3 objects (wicket keeper/spi bowler/fast bowler) ca be chose from 8 objects (8 remaiig cricket players). 3 Multiply the two results ( ad situatio as we wish to have a captai ad vice captai ad a wicket keeper, spi bowler ad fast bowler). Number of differet ways 0 P 8 P 3 90 336 30 40 exercise D Permutatios usig P r DiGiTal DoCS doc-98 SkillSHEET. Calculatig P r doc-986 Combiatorics We Evaluate: a 6 P 4 b 8 P c 9 P 3 d 4 P 4 e P f 3 P g 4 P + P h 8 P 6 7 P 3 i 6 P 3 P 4 j 3 P 4 P k 00 P 4 l 00 P 3 We A committee comprisig a presidet, vice-presidet, secretary ad treasurer is to be selected from a group of people. How may differet committees are possible? 3 I how may ways ca a first ad secod prize be give to lottery wiers? 4 We3 How may umbers greater tha 00 ca be formed from the digits, 3,, 7 ad 9 if a digit caot be used more tha oce? Joh has a -cet coi, a 0-cet coi, a 0-cet coi ad a $ coi. a I how may ways ca the cois be placed i a row? b I how may ways ca cois or 3 cois be chose if the order is take ito accout? 6 mc A magic pait set cotais seve magic colours that whe applied to paper produce other colours. The colour obtaied depeds o the order i which the colours are applied, ad at least two colours must be used. The umber of differet colours that ca be produced is: a 60 b 0 4 C 4 0 D 40 e 3 69 7 We4 A captai ad vice-captai are to be selected from a team of 8 footballers. From the remaiig 6, four players will be selected to be the full-back, full-forward, cetre-half back ad cetre-half forward. Calculate the umber of ways the 6 positios ca be allocated. 8 The Souther Belle s trai crew cosists of drivers ad 4 egieers. Each perso performs differet tasks. The drivers are chose from 6 available drivers ad the 4 egieers from 0 egieers. How may permutatios of the trai s crew are possible? 9 Three studets are to be chose from a group of 8 studets to fill the positios of school presidet, vice-presidet ad treasurer. After these appoitmets are made, more studets will be selected from the group to serve as secretary ad assistat secretary. Determie how may differet committees are possible. 0 A ovelty sports day carival ivolves 0 competitors. A prize is give to the wier of the first race, who the caot take part i the remaider of the races. The wier ad ruer-up of the secod race are awarded prizes ad are the elimiated from the remaider of the evets. Similarly, the first three place-getters of the third race are give prizes ad must drop out of the competitio. This is cotiued util the umber of competitors remaiig is the same as the umber of prizes to be awarded. How may differet ways ca prizes be awarded? 40 Maths Quest Mathematical Methods CAS
There are three separate budles of readig material comprisig 4 comics, ovels ad 3 magazies. They are placed together to form oe pile. a I how may ways ca this be doe if there are o restrictios o where idividual items are to be placed? b Determie the umber of permutatios if the order of the comic books i each budle does ot chage. e Permutatios ivolvig restrictios idetical objects So far our study of permutatios has bee based o the assumptio that the objects arraged were all differet (distiguishable). We will ow examie the situatio whe some of the objects are idetical (idistiguishable). A scrabble player has the followig letter tiles: A, A, A, B, C, D, E. If the As were distiguishable, we might cosider them to be A, A, A 3 ad could begi to list the possible arragemets of the 7 letters as follows: A A A 3 BCDE......... ad o s o A A 3 A BCDE......... ad o s o A A A 3 BCDE......... ad o s o A A 3 A BCDE......... ad o s o A 3 A A BCDE......... ad o s o A 3 A A BCDE......... ad o s o iteractivity it-07 Permutatios ivolvig restrictios Without listig them all, we ca calculate there are 7 P 7 7! 040 possible arragemets. But the As are ot distiguishable. So, really, the arragemets listed above are all the same as AAABCDE, which couts as oe arragemet. Because there are 3 A s we have 3! 6 times too may arragemets, hece we eed to divide 040 by 6. ChaPTer Combiatorics 4
7P This meas there are oly 7 7! 040 840 differet arragemets or permutatios of 3! 3! 6 7 objects where 3 of them are idetical. I geeral, the umber of arragemets of objects, p of which are idetical, is give by! p! Extedig this formula we have: The umber of ways of arragig objects that iclude p idetical objects of oe type, q idetical objects of aother type, r idetical objects of yet aother type ad so o is:! pqr!!!... WorkeD example I how may ways ca 4 idetical red marbles ad 3 idetical blue marbles be placed i a row? There are 4 + 3 7 objects altogether. The umber of ways the blue marbles ca be arraged is 3!, ad the umber of ways the red marbles ca be arraged is 4!. 3 Substitute the values ito the formula. Number of ways 7! 4! 3! 3 Grouped objects I how may ways ca the letters A, B, C, D be positioed i a row? We kow that this ca be doe i 4! ways, but what would be the aswer if the questio had bee: I how may ways ca the letters A, B, C, D be positioed i a row if A ad B must be ext to each other? The umber of arragemets will clearly be less tha 4! because of the restrictio imposed o A ad B. The figure below shows the 4! possible arragemets of A, B, C, D that iclude the ways A ad B are together. A B C D B A C D C B A D D B C A A B D C B A D C C B D A D B A C A C B D B C A D C A B D D C B A A C D B B C D A C A D B D C A B A D B C B D A C C D B A D A B C A D C B B D C A C D A B D A C B If A ad B are to be together, we cosider the problem to be oe of arragig 3 objects, say X, C ad D, where oe of the objects, X, is the group cotaiig A ad B. The figure below shows that there are 6 arragemets with A ad B together. A B C D C A B D C D A B A B D C D A B C D C A B The 3 objects ca be arraged i 3! ways, ad withi the group A ad B ca themselves be arraged i! ways (amely AB ad BA). The multiplicatio priciple is ow used so that the umber of arragemets whe A ad B are together is 3!!. Now cosider the permutatios if A, B, C must be together. Agai, we view the letters as cosistig of two objects, X ad D, where X is the group of letters A, B ad C. Thus we have two objects to arrage i! ways as show below. X D D X 4 Maths Quest Mathematical Methods CAS
Amog themselves the letters A, B, C cotaied i X have 3! differet arragemets as show below. A B C D A C B D B A C D B C A D C A B D C B A D D A B C D A C B D B A C D B C A D C A B D C B A Therefore the total umber of arragemets whe A, B ad C are together is! 3!. We ca geeralise this approach to iclude ay umber of groups of objects that are required to be together. If objects are to be divided ito m groups with each group havig G, G, G 3,... G m objects respectively, the umber of arragemets is give by m! G! G! G 3!... G m! WorkeD example 6 The letters of the word TABLES are placed i a row. How may arragemets are possible if the letters T, A ad B must be together? Cosider the letters T, A ad B as oe object (group). There are 4 objects to be arraged, amely the TAB group ad the letters L, E ad S. Idetify m ad G, G, G 3, G 4. 3 Apply the formula: m! G! G! G 3! G 4! G {T, A, B} G {L} G 3 {E} G 4 {S} m 4 Number of arragemets m! 3!!!! 4! 3! 44 WorkeD example 7 Five cars a Toyota, a Ford, a Holde, a Mazda ad a BMW are to be parked side by side. I how may ways ca this be doe if the Toyota ad BMW are ot to be parked ext to each other? The five cars ca be arraged i! ways without restrictio. Calculate the umber of arragemets where the Toyota ad BMW are together (4!!). (m 4, G, G, G 3, G 4 ) 3 Subtract from the urestricted umber of arragemets the umber of ways the two cars are together. Number of ways of arragig cars! Number of ways where the Toyota ad BMW are ot together! 4!! 0 48 7 WorkeD example 8 The letters of the word REPLETE are arraged i a row. I how may ways ca this be doe if the letters R ad P must ot be together? Fid the umber of urestricted arragemets of the 7 letters ad cosider that there are 3 idetical Es. TUTorial eles-47 Worked example 8 Number of ways of arragig 7 letters with 3 Es 7! 3! ChaPTer Combiatorics 43
Calculate the (restricted) umber of ways R ad P are together. Cosider R ad P as oe object so there are 6 objects to arrage. There are three Es to cosider (3! ways). R ad P ca be arraged i! ways withi their group. (m 6, G, G G 3 G 4 G G 6 ) 3 Subtract the umber of ways with R ad P together from the total umber of arragemets. Number of ways where R ad P are ot together 7! 6!! 3! 3! 840 40 600 exercise e Permutatios ivolvig restrictios We I how may ways ca idetical white beads ad 4 idetical yellow beads be arraged i a straight lie? Three -cet cois, two 0-cet cois ad six 0-cet cois are to be placed side by side. Determie how may ways this ca be doe. 3 mc The umber of permutatios usig the letters of the word LOOPHOLE is: a 30 b 3360 C 4000 D 40 e 3840 4 The toy set show i the photo cosists of a umber of idistiguishable brow horses, white horse, a cowboy ad 3 idistiguishable black horses. I how may ways ca they be placed ed to ed? How may differet 6-digit umbers ca be obtaied usig the digits 4, 6, 7, 6, 6 ad 4? 6 A party-light kit cosists of 0 coloured globes coected to each other i a straight lie. a If there are red globes, 6 blue globes, 7 yellow globes ad a umber of gree globes as show at right, fid how may differet arragemets of coloured globes are possible. b How may differet permutatios of coloured globes are there if the first ad last globes must both be red? 7 We6 Fid how may arragemets are possible altogether whe the letters of the word CHAIR are placed i a row ad C ad H are to be ext to each other. 8 The digits, 3, 6, ad 7 are used to make a -digit umber. How may differet umbers are possible if the digits 3, ad 7 must be together? 9 Maria, Steve, James, Sofia, Ni ad Alfredo are stadig ext to each other. Calculate how may ways this ca be doe if Maria ad James are ot to stad ext to each other. 0 We7 Establish the umber of ways i which 7 differet books ca be placed o a bookshelf if particular books must occupy the ed positios ad 3 of the remaiig books are ot to be placed together. mc Te athletes lie up for a race. The umber of permutatios whe three of the athletes Sam, Troy ad Pablo would be ext to each other is: a 3 68 800 b 09 600 C 4 90 D 443 00 e 4 838 400 A carpeter has 3 idetical hammers, differet screwdrivers, idetical mallets, differet saws ad a tape-measure. She wishes to hag the tools i a row o a tool rack o the wall. I how may ways ca this be doe if the first ad last positios o the rack are to be mallets ad the hammers are ot to be all together? 44 Maths Quest Mathematical Methods CAS
3 We8 Decide i how may ways the letters of the word ABRACADABRA ca be arraged i a row if C, R ad D are ot to be together. 4 mc The umber of ways the letters of the alphabet ca be placed i a straight lie with the restrictio that the letters of the setece UP THE BIG SKY WORLD must ot be together is: a 6!!6! b 6! + 6! C 6! 8! D 6!6! e 6!6!6! F arragemets i a circle Aa, Betty ad Li stad o the circumferece of a circle paited o the school s playgroud. I how may differet arragemets ca the three girls stad? The figure below shows the two arragemets for the girls positios o the circle. Aa Aa Betty Li Li Betty Notice that Aa is locked i positio to provide a referece poit, ad Betty ad Li are arraged aroud Aa i! ( ) ways. Compare this with the 3! ( 6) arragemets i a lie. ABL BAL BLA LBA ALB LAB (A is Aa, B is Betty, L is Li) Susie ow jois the group to make 4 people i a circle. We ca desigate ay of the 4 girls i the circle as our start by fixig oe perso (i this case, Aa) i oe positio ad arragig the remaiig girls aroud her. This reduces, by oe perso, the umber of girls to arrage. A A A A A A B L B S L B L S B L S B S L S B S L (A is Aa, B is Betty, L is Li, S is Susie) There are 3! ( 6) ways of arragig 4 people i a circle. Compare this with 4! ( 4) arragemets i a lie. I geeral: distiguishable objects ca be arraged i a circle i ( )! ways. I how may ways ca these five childre be arraged i a circle? ChaPTer Combiatorics 4
WorkeD example 9 I how may ways ca the vowels of the alphabet be arraged i a circle? The vowels are a, e, i, o, u. Therefore, there are objects to arrage. Use ( )! with. Number of ways ( )! ( )! 4! 4 WorkeD example 0 Calculate the umber of arragemets i a circle that are possible usig the letters of the word UNUSUALLY. There are 9 letters, so use 9 with ( )! 9 We eed to cosider repetitio of letters. There are three Us ad two Ls. (9 )! Number of arragemets 3!! 8! 3!! 3360 WorkeD example I how may ways ca 6 people sit aroud a table if two particular people must be seated ext to each other? Cosider the two people required to sit together as beig oe object. So there are objects to arrage i a circle. The two people ca be arraged i! ways. TUTorial eles-48 Worked example 3 Use the multiplicatio priciple. Number of ways ( )!! 4!! 48 exercise F arragemets i a circle We9 Calculate the umber of ways i which the letters of the word PENCIL ca be arraged i a circle. mc Eight childre hold hads to form a circle i the playgroud. The umber of ways this ca be doe is: a 680 b 400 C 3680 D 430 e 040 3 We0 Determie the umber of arragemets i a circle that are possible whe the letters of the word EXCELLENT are used. 4 A child uses coloured dots o paper to represet the hour marks of a clock face. How may permutatios are possible if there are 4 orage dots, white dots, black dots ad purple dot? 46 Maths Quest Mathematical Methods CAS
We A family of 3 adults, 3 boys ad 3 girls are sittig aroud a circular dier table. Fid the umber of seatig positios that are possible if the 3 boys are to be together. 6 A special pizza cosists of 0 slices with differet toppigs used. If slices are Capricciosa, slices are Supreme ad 3 slices are Ham ad Pieapple, how may differet arragemets of pizza slices are possible? 7 A maufacturer of merry-go-rouds uses 8 idetical woode horses, 4 idetical plastic motorbikes ad differet miiature cars. They are all equally coected aroud the rim of a circular movig base. Establish how may differet arragemets there ca be if the cars are ot to be placed i cosecutive positios. 8 mc Te owers of pedigree dogs will eter the area to parade their dogs by walkig aroud a circular track. Ufortuately, 3 particular dogs caot get alog together ad so caot parade if all 3 are ext to each other. There appears to be o problem if ay two of this group of 3 dogs are together. The umber of ways of avoidig this problem is: a 38 848 b 387 07 C 36 880 D 33 640 e 34 06 9 I how may ways ca the letters of the word POTATOES be arraged i a circle? 0 mc The letters of the word FULFILLED are to be arraged i a circle. The umber of arragemets possible whe U ad E are together or whe U, E ad D are together is: a 340 b 940 C 000 D 00 e 80 To publicise a veue, a hotel maager gave a gift to each of promiet busiesspeople as they wet ito the coferece room ad seated themselves at a roud table to begi discussios. The gifts comprised 4 foutai pes, pocket electroic orgaisers ad 3 calculators. Calculate what fractio of the possible urestricted arragemets is the umber of arragemets that has 4 busiesspeople who have bee give a foutai pe sittig ext to each other. DiGiTal DoC doc-987 WorkSHEET. G Combiatios usig C r Takig combiatios ivolves the selectio of r objects from objects without cosideratio for the order of the elemets. For example, the umber of permutatios of two letters selected from the letters A, B, C, D is 4 P. The arragemets are: AB AC AD BC BD CD BA CA DA CB DB DC If we are ot cocered with order, there are oly 6 selectios: AB AC AD BC BD CD The! ways of arragig the elemets of the -elemet subgroup are ot cosidered. Now cosider the selectio of 3 letters from A, B, C, D. The umber of ordered subsets is 4 P 3, ad each subset of 3 elemets ca be arraged i 3! ways. Therefore 4 P 3 is the umber of uordered subsets of 3 objects multiplied by the umber of ways the 3 objects ca be arraged. I geeral terms it ca be stated that P r is the umber ( C r ) of uordered groups of r objects multiplied by the umber of arragemets (r!) of r objects. That is, P r C r r! so that P C r r r!. Now by the defiitio of! P r we have: ( r)!!! Cr r! ( r)! r!( r)! The umber of combiatios is usually deoted by Cr or r, so we have:. The umber of combiatios of r objects selected from objects is: where, r are atural umbers ad r.. r P C! or r C r r r r P r r!! Cr r!( r)! ChaPTer Combiatorics 47
The fuctio C r is a stadard mathematical fuctio to be foud o scietific, graphics ad CAS calculators. Special cases. If r 0, the C!! r C0. 0! ( 0)!! This implies that there is oe way of selectig 0 objects from objects.. If r, the C!! r C.! ( )!! 0! There is oe combiatio of objects take from objects. 3. If r, the! ( )! C r C.! ( )! ( )! If objects are take oe at a time from objects, there are combiatios. From cases ad we coclude that C 0 C. This is a istace of the geeral case that: C r C r or r r For example, 7 7! C4 4!3! ad 7 7! C3 3!4! so 7 C 4 7 C 3 WorkeD example Evaluate 0 C 3. Use the defiitio C r!!( r)!. Alteratively, use the combiatios feature of the CAS calculator with 0 ad r 3. 3 Record the result. 0 0 0! C3 3!(0 3)! 0! 0 9 8 7! 0 9 8 3! 7! 3! 7! 3 0 C r (0, 3) WorkeD example 3 Evaluate 00 98. 00 Express 98 i factorial form. 00 00! 98 98!! 00 99 98! 98! 00! 00 99 98! 3 Evaluate. 00 99 490 48 Maths Quest Mathematical Methods CAS
WorkeD example 4 I how may ways ca a committee of boys ad 3 girls be formed from a group cosistig of boys ad 8 girls? Select boys from boys. Select 3 girls from 8 girls. 3 Use the multiplicatio priciple ( ad situatio). Number of ways C 8 C 3 0 6 60 WorkeD example A committee of 6 is to be formed from a group of me ad 4 wome. a How may committees ca be formed? b How may committees cotai 3 me ad 3 wome? c How may committees cotai at least 4 me? TUTorial eles-49 Worked example a Use 9 ad r 6 with C r. a Number of committees 9 C 6 84 b Select 3 me from me ad 3 wome from 4 wome. Use the multiplicatio priciple. ( ad situatio) c At least 4 me meas 4 me ad wome or me ad woma. b Number of committees C 3 4 C 3 0 4 40 c Number of committees C 4 4 C + C 4 C Select 4 me ad wome. 6 + 4 3 Select me ad woma. 30 + 4 4 Sum the aswers because the evets are mutually exclusive ( or situatio). 34 exercise G Combiatios usig C r We Calculate each of the followig. a C b 4 C 3 c 6 C d 8 C 0 e 9 C 9 Evaluate each of the followig. a 6 4 b 7 c 0 3 We3 Determie the value of each of the followig. a 30 9 b 3 c 64 6 d d 9 3 38 34 e 6 e 9 4 4 mc The value of 4 C + 3 C 3 is: a 4 b 90 C 80 D 94 e 70 Calculate each of the followig. a 3 C ad 3 C b 4 C ad 4 C 3 c C ad C 3 d 9 C 3 ad 9 C 6 DiGiTal DoC doc-986 Combiatorics ChaPTer Combiatorics 49
6 Copy ad complete the followig. a 0 C 7 0 C b 00 C 9 00 C 7 I how may ways ca objects be chose from? 8 How may combiatios are possible if umbers are chose from 6 i a mii-lotto game? 9 A studet must choose types of party food from the followig list: sausage rolls, potato crisps, fairy bread, party pies, cheezels, cocktail frakfurts ad celery sticks. How may differet combiatios may be chose? 0 A committee of 6 must be chose from a meetig of 30 people. How may differet committees are possible? We4 I how may ways ca a group of 3 boys ad 4 girls be formed from a group cosistig of 4 boys ad 6 girls? A magazie pile i a waitig room cotais 6 glamour magazies ad 7 computer magazies. I how may ways ca a patiet choose glamour ad 3 computer magazies to flick through durig a legthy wait? 3 A school offers 0 sciece subjects ad humaities subjects to prospective Year studets. I how may ways may a studet choose 4 sciece ad humaities subjects? 4 How may 0-card hads cotaiig exactly 7 hearts ad 3 spades are possible from a stadard -card deck? We A committee of parets is to be established from a group of 6 me ad 4 wome. a Fid how may differet committees ca be formed. b How may differet committees are possible cosistig of 3 me ad wome? DiGiTal DoC doc-988 SkillSHEET. listig possibilities 6 A school orgaises a adveture camp for its Year studets, who must choose or 3 activities from the followig: paraglidig, abseilig, skydivig ad bugee jumpig. I how may ways may a group of activities be chose? 7 A ice-cream vedor offers chocolate, strawberry ad vailla ice-creams with oe, two or three scoops. How may differet ice-creams are possible? (Assume that you caot choose two scoops of the same flavour for ay oe oe ice-cream.) 8 A basketball squad of 0 must be chose from a group of 8 wome ad 6 me. How may squads are possible: a without restrictio? b if the squad cotais 6 wome ad 4 me? c if the squad must cotai at least 6 wome? d if the squad cotais all the me? 9 A sub-committee of 3 people must be chose from a group of 9 teachers (which icludes the pricipal). How may sub-committees may be chose: a that cotai the pricipal? b that do ot cotai the pricipal? DiGiTal DoC doc-989 Ivestigatio Pascal s triagle 0 To wi LottoMaia, the umbers etered o the player s etry ticket must be the same as umbers that are radomly selected from the umbers to 30. a How may differet etries are possible? b What is the percetage icrease i the umber of possible combiatios if the umbers are radomly selected from the umbers to 3? mc A paiter has 7 colours at her disposal. The umber of additioal colours that ca be obtaied by mixig equal amouts of ay umber of the 7 colours is: a 00 b 8 C 040 D 0 e 0 Determie the umber of ways i which 8 people ca be divided ito equal groups. 3 mc The umber of ways i which 0 objects ca be divided ito uequal groups is: a 38 b 83 C 90 D 640 e 6 0 Maths Quest Mathematical Methods CAS
h applicatios to probability We defie the probability of a evet to be: umberoffavourable outcomes Pr(evet) total umber of possible outcomes The methods we have used to calculate permutatios ad combiatios ca also be applied to problems ivolvig probability. WorkeD example 6 Romia makes a guess as to which of 0 swimmers will come first ad secod i a race. What is the probability that her guess will be right? Calculate i how may ways swimmers ca be chose from 0 swimmers, where the order is take ito accout. Use P r where 0 ad r. Use the formula for probability. The umber of favourable outcomes is because Romia makes oly oe guess. Pr (correct guess) 0 P 90 WorkeD example 7 A computer radomly iterchaged the letters of the word CREATIONS. Fid the probability that the letters A ad T ed up together. If A ad T are together, treat them as oe object; therefore, we have 8 objects AT ca be arraged i! ways. 3 Use the formula for probability to fid the umber of ways the 9 letters ca be arraged (total umber of possible outcomes). Pr (A ad T are together) 8!! 9! 9 WorkeD example 8 A committee of people is to be formed by choosig members from a group of 6 me ad 4 wome. What is the probability that the committee will cosist of 3 me ad wome? Calculate the umber of ways i which 3 me ca be selected from 6 me ad wome ca be chose from 4 wome. Use the multiplicatio priciple to establish the umber of favourable outcomes of the committees cosistig of 6 me ad 4 wome. 6 C 3 ad 4 C 6 C 3 4 C ChaPTer Combiatorics
3 Usig the formula for probability, determie the umber of ways i which people ca be selected from the group of 0 people (total umber of possible outcomes). Pr (3 me ad wome) 6C 3 4C 0C 0 6 0 WorkeD example 9 Eight people radomly seat themselves about a circular table. What is the probability that 3 particular people will be sittig ext to each other? Treat the 3 people as oe object; therefore, there are 6 objects to arrage. Use the formula ( )! for arragemets i a circle for the situatio where the 3 people are together. 3 Usig the formula for probability, calculate the total umber of possible outcomes for the 8 people, usig ( )! 3! (6 )! 3! Pr (3 particular people seated together) (6 )! 3! (8 )!! 3! 7! 70 040 0.43 WorkeD example 30 Two bags (A ad A ) cotai blue marbles (B) ad other coloured marbles (B ). A bag is radomly selected, the from that bag a marble is radomly selected. The table below describes the distributio of marbles betwee the bags. Bag A Bag A TUTorial eles-460 Worked example 30 blue marbles 4 blue marbles 3 other marbles 6 other marbles a What is the probability of choosig bag A ad the a blue marble? b What is the probability of ot choosig bag A ad the obtaiig a blue marble? c What is the probability of choosig a blue marble? umber offavourable outcomes a Fid the probability of choosig bag A. a Pr(A) total possible outcomes C C Maths Quest Mathematical Methods CAS
Fid the probability of choosig a blue marble from bag A. Pr (B froma) umberoffavourable outcomes total possible outcomes C 8C 8 3 Fid the probability of choosig bag A ad the a blue marble. Pr (A B) 8 6 b Fid the probability of choosig bag A. b Pr (A ) umberoffavourable outcomes total possible outcomes C C Fid the probability of choosig a blue marble from bag A. 3 Fid the probability of choosig bag A ad the a blue marble. c A blue marble ca be selected from bag A or bag A. Pr (B froma ) umberoffavourable outcomes total possible outcomes 4 0 Pr (A B) c Pr (B) + 6 4 80 C C Pr ( A B) Recall from chapter that, for coditioal probability, Pr ( A B), Pr (B) 0. Rearragig this formula gives Pr ( B) From Worked example 30 above, otice that Pr (A B) Pr (A B)Pr (B) [] Pr (B) Pr (A B) + Pr (A B) or Pr (A) Pr (A B) + Pr (A B ) [] Combiig the iformatio from equatios [] ad [], we have Pr (A) Pr (A B)Pr (B) + Pr (A B ) Pr (B ) This expressio is kow as the Law of Total Probability ad was briefly discussed i chapter. Aother way to visualise this rule is to use a tree diagram. The tree diagram below shows the situatio described i worked example 30. Notice how the probability of selectig a blue marble from bag A is deoted as Pr (B A). This is because the probability of selectig a blue marble from bag A is coditioal o selectig bag A to begi with. A 8 3 8 (B A) (B' A) 8 Pr (A B) 3 8 Pr (A B') 6 3 6 A' 4 0 6 0 4 0 6 0 (B' A' ) Pr (A' B') 0 (B A' ) Pr (A' B) 3 0 ChaPTer Combiatorics 3
So, worked example 30 could also have bee solved usig a tree diagram or the Law of Total Probability. WorkeD example 3 The probability that Suzae will pass her examiatio give that she had help from her tutor is. The probability that Suzae does ot pass her exam give that she did ot see her tutor is. If the probability of Suzae seeig her tutor is, what is the probability of her passig her exam? /DraW Defie T ad E. Write dow all the iformatio that is give i the questio. Draw a tree diagram to represet the iformatio. 3 Usig the formula Pr (E) Pr (E T ) Pr (T ) + Pr (E T ) Pr (T ), substitute all the kow values from the tree diagram. Let T havig help from the tutor. Let E passig the exam. Pr ( E T),Pr( E T ), Pr ( T) T T 4 3 E T E T E T E T Pr ( E) 3 3 4 Iterpret the result. The probability that Suzae will pass her exam is 3. DiGiTal DoC doc-986 Combiatorics exercise h applicatios to probability We6 Jey, Haka ad Miriam are competig i a car race agaist other drivers. Their fried Mary predicts that they will cross the fiish lie first, secod ad third respectively. What is the probability that Mary is right? We7 The letters of the word PRODUCE are radomly reordered. Calculate the probability that the letters P ad E will be together. 3 We8 Six people selected from me ad 7 wome are to form a committee. Work out the probability that the committee will cosist of 3 me ad 3 wome. 4 mc The letters A, B, C, D, E ad F are radomly placed i a row. The probability that the letters A ad B will occupy the first ad secod positios respectively is: a b 3 C 30 Six cards are radomly distributed from a stadard pack of playig cards. Determie the probability that exactly oe of the 6 cards is a quee. 6 From a toy set cosistig of 4 dolls ad clows, toys are chose at radom. Fid the probability that the toys are clows or dolls. 7 mc From a group of 3 childre ad 8 adults, will be chose to receive prizes. The probability that childre ad 3 adults will be awarded a prize is: a 3 8 3 b 3 8 3 8 C 3 + 8 3 D 6 D 3 8 3 e e 3 3 8 3 3 + 8 8 We9 A group comprisig 6 people is sittig aroud a table. Fid the probability that two particular people are sittig ext to each other. 4 Maths Quest Mathematical Methods CAS
9 Te people are seated at a circular diig table. Fid the probability that two particular people will be sittig ext to each other. 0 mc Six mothers ad their 6 daughters radomly arrage themselves i a circle. The probability that Susa is ext to her daughter Jeaette is: a b C D e 4 6 3 Four letters are radomly selected from the word ENCYCLOPAEDIA. Fid the probability that oe letter E will occur i the selectio of 4 letters. A school captai ad vice-captais are to be chose from a group of boys ad 6 girls. What is the probability that all 3 positios will be take by: a boys? b girls? c two boys ad oe girl? d at least two girls? 3 Four colours are radomly picked from the 7 differet colours of the raibow. Calculate the probability that yellow will ot be oe of the colours chose. 4 A dealer draws three cards from a deck of cards. What is the probability that she draws: a o quees? b at least quees? c exactly oe heart? Five letters are radomly selected from the letters of the word HOLIDAYS ad placed i a row. Calculate the probability that the first letter chose is a cosoat. 6 mc Iside a box are objects of which m are white. If r objects are radomly take out of the box ad placed i a row, the probability that the first object is white is: a m b m! C m D m+! e m 7 mc A -digit umber is radomly formed usig the digits,, 3, 4,, 6, 7, 8 ad 9. If a digit caot be used more tha oce i the umber: a the probability that the umber is eve is: 9 7 4 3 a b C D 00 9 9 78 b the probability that the umber is betwee 30 000 ad 0 000 is: a 0 7 b 9 73 C 9 8 A debatig team of 6 people is to be formed from a group cosistig of males ad 6 females. a What is the probability that the team will cosist of at least oe male? b What is the probability that the team will have at least four females? 9 We30 Two small crates (X ad Y) cotai apples (A) ad baaas (B). A crate is radomly selected, the from that crate a piece of fruit is X Y radomly selected. The table at right describes the distributio of fruit betwee the crates. 6 apples 4 apples a What is the probability of selectig crate X ad from it, a baaa? b What is the probability of selectig crate Y ad from it, a baaa? c What is the probability of selectig a baaa? baaas 7 baaas d Fid the probability of selectig a baaa usig Pr (B) Pr (B X)Pr (X) + Pr (B Y)Pr (Y). 0 Give Pr (B A) 3, Pr (B A ), ad Pr (A) 3, fid Pr (B) usig the Law of Total Probability. 3 4 We3 The probability that Tim is late for school is, but he has a exam o Friday. The chace of 3 him passig his exam give that he is o time to school is. If he is late, his chace of ot passig the 7 exam is. What is the chace that Tim will pass his exam? Elei loves chocolates. She particularly loves soft-cetred chocolates. She is offered a box of chocolates to select from, but all the chocolates are wrapped. The probability of selectig a softcetred chocolate give that it is dark chocolate is, ad the probability of selectig a hard cetre give that it is milk chocolate is 4. If there are 7 milk chocolates i the box, fid the probability of selectig a 7 soft-cetred chocolate. D 3 78 e e 4 63 4 63 ChaPTer Combiatorics
3 Fred s chace of beig selected for the soccer team this seaso is 8. The probability of Fred goig o the school trip give that he is selected for the soccer team is 7, whereas the chace of him ot goig o the school trip give that he is ot selected for the soccer team is 3. What is the probability 4 that Fred will go o the school trip? 4 The chace of a spriter wiig a race give that his archrival rus is. If his archrival does ot ru, 3 the spriter has a chace of wiig. His archrival is ijured ad has a chace of ruig at all. 8 4 Use the Law of Total Probability to fid the probability that the spriter wis the race. 6 Maths Quest Mathematical Methods CAS
Summary The additio ad multiplicatio priciples Permutatios Factorials arragemets i a circle Combiatios usig C r The additio priciple states that if two operatios ca be performed i A or B ways respectively, the both operatios ca be performed together i A + B ways. The multiplicatio priciple states that if two operatios ca be performed i A ad B ways, the both operatios ca be performed i successio i A B ways. A permutatio is the arragemet of objects i a defiite order. The multiplicatio priciple is commoly used i calculatig the umber of possible permutatios.! P r ( r)! The umber of ways of arragig objects that iclude p idetical objects of oe type, q idetical! objects of aother type, r idetical objects of yet aother type ad so o is: p! q! r! objects divided ito m groups, with each group havig G, G, G 3,... G m objects respectively, has m! G! G! G 3!... G m! arragemets. The factorial of a positive whole umber is defied as:! ( ) ( ) ( 3)... 3 with 0!! ( )! ( ) ( )! ad so o. distiguishable objects ca be arraged i a circle i ( )! ways. The same methods are applicable to arragemets i a circle as the methods used for idistiguishable objects whe there are restrictios o the possible arragemets. The umber of combiatios whe r objects are selected from objects is deoted by C r or r. C r r! P r or r!( r)! r! C r C r applicatios to probability umberoffavourable outcomes The probability of a evet: Pr (evet) total umberof possible outcomes The Law of Total Probability states: Pr (A) Pr (A B)Pr (B) Pr (A B )Pr (B ) ChaPTer Combiatorics 7
Chapter review ShorT aswer There are 7 airlies that have flights from Australia to Sigapore, 6 airlies that offer flights from Sigapore to Europe, ad airlies that service the route from Europe to America. Determie the umber of differet travel arragemets possible to get from Australia to America via Sigapore ad Europe. Seve people form a queue to board a bus. How may differet queues are possible? 3 The digits 3,, 6 ad 8 are used to form umbers greater tha 00. If a digit may be used oce oly ad ot all digits have to be used, how may differet umbers ca be formed? 4 Seve differet books are to be placed o a shelf. If a particular book must occupy the first positio, fid the umber of permutatios possible. I how may ways ca first, secod ad third prizes be awarded to people competig i a maratho? 6 A team of at least people must be chose from a group of moutaieers to mout a rescue missio. How may differet teams may be chose? multiple ChoiCe Samatha ca get to work by walkig, by takig her car or by usig public trasport (trai, tram, bus or taxi). The umber of differet ways she ca get to her work is: a 3 b C 4 D 6 e Malcolm is guessig someoe s house umber. He kows that the umber is a odd umber ad is betwee 30 ad 60. Assumig that the same guess is ot made twice, the maximum umber of guesses he ca make is: a b 0 C 30 D 4 e 3 The total umber of -digit, 3-digit ad 4-digit odd umbers that ca be formed usig the digits 6, 4,,, whe a digit caot be used more tha oce is: a 00 b 80 C 70 D 0 e 8 4 The value of 998! 996! is: a udefied b 000 999 C 996! D 998 997 e 998 997 996 The value of 9! 7! is equivalet to: a 7 7! b! C 7! 9 D 8! e 7 8! 6 The value of 7 P is: a b 4 C 0 D 008 e 040 7 The umber of permutatios usig the letters of the word MISSISSIPPI is: a 4! b!! C D!!4!4! e 4!4!! 4! 8 Five letters are chose from the letters of the word WATERING ad placed i a row. The umber of ways i which this ca be doe if the last letter is to be W is: a 840 b 0 C 0 D 40 e 6 9 A family cosistig of a mother, father, 3 sos ad 4 daughters lies up for a photograph. How may ways ca this be doe if the daughters must be together? a 9! b 6!4! C!4! D!3!4! e 0! 0 Eleve members of a cricket team are to be seated i a circle. The umber of possible arragemets is: a! b 0! C! D! 0! 0! e The letters of the word MUSICAL are to be arraged i a circle. If the letters U ad S must ot be together, the umber of possible arragemets is: a 480 b 78 C 440 D 3600 e 038 8 Maths Quest Mathematical Methods CAS
Joaa has decided to study at uiversity. Her course requires that she udertake at least subjects for the year. If 4 subjects are beig offered, the umber of subject combiatios is: a 36 b 4 C D e 0 3 Four pieces of fruit are selected from a box cotaiig orages ad 6 apples. The umber of selectios that cotai at least orages ad apple is: a 0 b 0 C 60 D 90 e 0 4 Five letters are radomly selected from the word ENERGISE. The probability that the letter E will appear i the group of letters is: a 8 b 6 C 3 8 D 8 e 6 A 3-, 4- or - digit umber is to be formed usig digits take from 8, 4, 3, 6 ad 7. If a digit may be used more tha oce, how may differet umbers ca be made? The 4 fastest ruers i a race will qualify for the fials. If there are competitors, determie the umber of differet ways i which the race ca fiish. 3 Evaluate 9! + 8! 6! + 3! 4 Fid the umber of ways the letters of the word ARRANGEMENT ca be placed i a row. Aa, Belida, Chie, Deaa ad Erica are liig up for cocert tickets. If Belida ad Deaa do ot wat to be ext to each other, what is the umber of possible queues? 6 Te childre are arragig themselves i a circle. Calculate the umber of ways this ca be doe if three particular childre are ot to be ext to each other. 7 Two studets from a group of 8 studets are to be class captai ad vice-captai. From the remaiig cadidates, two will become class moitors. Fid the umber of ways this ca be doe. 8 A class cosists of 4 studets. If a iitial group of 4 must be chose to go for a measles ijectio, how may differet combiatios may be selected for that group? 9 A committee of people is to be established usig members from a group of 6 me ad 7 wome. What is the probability that the committee will cotai me ad 3 wome? 0 The letters of the word FEATURING are radomly rearraged. Fid the probability that the letters of the word FEAT are together, though ot ecessarily i the order show. Two wome ad three me approach a ATM at the same time. a How may differet queues are possible if the positio of each perso i the queue is take ito accout? b How may queues of at least two people are possible if the positio of each perso i the queue is ot take ito accout? exteded respose ChaPTer Combiatorics 9
DiGiTal DoC doc-980 Test Yourself Chapter From a group of 0 female studets, female staff, 8 male studets ad 3 male staff, a committee of 6 is to be formed. a Fid the umber of differet committees if: i there are o restrictios ii all committee members must be studets iii oe female ad oe male staff member must be o the committee iv there is a equal umber of males ad females o the committee v oe particular studet must be o the committee vi oe particular studet must ot be o the committee vii the committee must comprise male staff members, male studets, female staff member ad female studet. b Fid the probability that: i oly studets are selected for the committee ii all the staff are selected for the committee iii exactly staff ad 4 studets are selected. 3 I the game of Tattslotto, a barrel cotais forty-five balls umbered to 4, of which eight are radomly draw. The first six of these umbered balls are the wiig umbers. The fial two draw are called supplemetary umbers. Whe you purchase a stadard ticket, you may select six umbers i each game. Prizes are awarded accordig to how may of your six umbers match those draw from the barrel. To wi the first prize (divisio oe), all six of your umbers must match the six wiig umbers draw from the barrel. To wi the secod prize (divisio two), five of your umbers must match the wiig umbers ad your remaiig umber must match oe of the supplemetary umbers. To wi the third prize (divisio three), five of your umbers must match the wiig umbers. (Your remaiig umber does ot match ay of the umbers draw.) a What is the probability of wiig divisio oe? b What is the probability of wiig divisio two? c What is the probability of wiig divisio three? d What is the probability of wiig at least a divisio three prize? 60 Maths Quest Mathematical Methods CAS
ICT activities Chapter opeer DiGiTal DoC 0 Quick Questios doc-983: Warm-up with te quick questios o combiatorics (page 9) a The additio ad multiplicatio priciples TUTorial We4 eles-44: Use the multiplicatio ad additio rules to calculate the umber of differet luches ad the umber of differet dishes that ca be ordered at a cafeteria (page 3) b Permutatios TUTorial We7 eles-4: Use permutatios to determie the umber of ways three awards ad two prizes ca be distributed to five differet people (page 34) C Factorials DiGiTal DoC WorkSHEET. doc-984: Determie the umber of combiatios i differet scearios ad calculate expressios ivolvig factorials (page 37) D Permutatios usig P r TUTorial We4 eles-46: Use permutatios to determie the umber of differet way five positios ca be determied from te people (page 39) DiGiTal DoCS SkillSHEET. doc-98: Practise calculatig P r (page 40). doc-986: Ivestigate combiatorics usig a spreadsheet (page 40) e Permutatios ivolvig restrictios iteractivity Permutatios ivolvig restrictios it-07: Cosolidate your uderstadig of permutatios ivolvig restrictios (page 4) TUTorial We8 eles-47: Determie the umber of ways the letters i a particular word ca be arraged if two specific letters caot be adjacet (page 43) F arragemets i a circle TUTorial We eles-48: Determie the umber of ways six people ca be arraged aroud a table, if two specific people must be seated ext to each other (page 46) DiGiTal DoC WorkSHEET. doc-987: Calculate permutatios ad evaluate expressios ivolvig P r (page 47) G Combiatorics usig C r TUTorial We eles-49: Calculate the umber of differet committees that ca be formed from a group of five me ad four wome, give three varyig costraits (page 49) DiGiTal DoCS doc-986: Ivestigate combiatorics usig a spreadsheet (page 49) SkillSHEET. doc-988: Practise idetifyig ad listig possible outcomes (page 0) Ivestigatio doc-989: Ivestigate Pascal s triagle (page 0) h applicatios to probability TUTorial We30 eles-460: Apply the law of total probabilities ad the probability of a evet to calculate probabilities of selectig specific coloured marbles from two bags (page ) DiGiTal DoC doc-986: Ivestigate combiatorics usig a spreadsheet (page 4) Chapter review DiGiTal DoC Test Yourself doc-980: Take the ed-of-chapter test to test your progress (page 60) To access ebookplus activities, log o to www.jacplus.com.au ChaPTer Combiatorics 6
Aswers CHAPTER CombiaToriCS exercise a The additio ad multiplicatio priciples 3 97 4 E 6 B 7 0 9 a 4 8 7 b 440 0 E D a E is Eglish, M is Mathematics, L is Laguage, S is Sciece. E, M, L, S, EM, EL, ES, ML, MS, LS, EML, EMS, ELS, MLS, EMLS b There are ways i total. 3 a Walk/walk, walk/bus, trai/walk, trai/ bus, bus/walk, bus/bus b Walk Walk Bus Trai Bus Walk Bus Walk Bus 4 6 30 exercise b Permutatios 4 70 3 336 4 00 000 7 a 64 0 b 6 6 40 30 8 30 9 a 70 b 80 0 80 30 30 3 C 4 C C 6 6 7 Lose $7 44.00 8 a i 479 00 600 ii 36 880 b 84 03 9 a 60 000 b I the th year exercise C Factorials a 4 b 36 880 c 479 00 600 d 8 e 96 f 438 g 67 a 0 b 6 c 880 d 0 e 40 06 f 77 98 g 3 376 h 439 08 448 3 B 4 D a b c 840 d 0 e 4 6 a 0 30 b 999 000 c 400 d 438 000 e 6 69 480 f 000 7 a b 76 c 970 d 999 000 e 3906 8 E 9 a 4 b 04 c 96 d 49 9 e 0 a 40 b 90 c 4 d 68 e 4 exercise D Permutatios usig P r a 360 b 6 c 04 d 4 e 637 600 f 6 g 7 h 9 90 i 0 j 36 k 94 09 400 l 7 880 400 303 600 3 0 4 300 a 4 b 36 6 E 7 3 366 080 8 00 9 670 0 3 68 800 a 36 880 b 70 exercise e Permutatios ivolvig restrictios 6 460 3 B 4 040 60 6 a 793 0 70 b 47 06 880 7 48 8 36 9 480 0 68 C 68990 3 78 64 4 A exercise F arragemets i a circle 0 E 3 3360 4 6930 430 6 7 44 8 D 9 60 0 D 4 6 exercise G Combiatios usig C r a 0 b 4 c 6 d e a b c 4 d 84 e 94 3 a 30 b 48 c 4664 d 73 8 e 8 7 4 A a 3, 3 b 4, 4 c 0, 0 d 84, 84 6 a 0 C 7 0 C 3 b 00 C 9 00 C 9 7 79 8 9 0 93 7 60 3 00 4 490 776 a b 0 6 0 7 7 8 a 00 b 40 c 9 d 70 9 a 8 b 6 0 a 406 b 8% E 70 3 A exercise h applicatios to probability 336 3 66 7 4 C 0.336 6 4 9 7 D 8 9 9 0 E 33 a 33 3 c 4 3 7 4 a 0.783 b 0.03 c 0.436 b 4 33 d 9 33 8 6 C 7 a C b C 46 8 a b 8 46 46 9 a b 7 c 6 8 0 3 4 96 ChaPTer review d 6 3 6 3 3 ShorT aswer 0 7! 040 3 48 4 6! 70 30 6 6 multiple ChoiCe D A 3 B 4 D A 6 C 7 C 8 A 9 B 0 B A D 3 A 4 E exteded respose 387 790 3 40 486 4 494 800 7 6 33 640 7 840 8 0 66 9 0.408 or 7 49 0 0.0476 or a 0 b 6 6 Maths Quest Mathematical Methods CAS
a i 6 096 44 ii 760 68 iii 44 890 iv 04800 v 80 668 vi 4 786 vii 8 360 b i 0.43 ii 0 (egligible) iii 0. 3 a.3 0 7 c.73 0 b.47 0 6 d.89 0 ChaPTer Combiatorics 63