Permutations and Combinations

 Christiana Cooper
 6 months ago
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1 13 Pemutatios ad Combiatios TERMINOLOGY Aagemets: Diffeet ways of ogaisig objects Combiatios: Aagemets of objects without eplacemet o epetitio whe ode is ot impotat. The otatio used is C fo selectig objects fom whee ode does t matte Factoial: A factoial is the poduct of cosecutive positive iteges fom dow to oe. Fo example 6! 6 x 5 x 4 x 3 x 2 x 1 Fudametal coutig piciple: If oe evet ca occu i p ways ad a secod idepedet evet ca occu i q ways, the the two successive evets ca occu i p x q diffeet ways Odeed selectios: Selectios that ae take i a paticula positio Pemutatios: The aagemet of objects without eplacemet o epetitio whe ode is impotat. The otatio used is P fo selectig objects fom whee ode mattes Radom expeimets: Expeimets that ae made with o patte o ode whee each outcome is equally likely to occu Sample space: The set of all possible outcomes i a evet o seies of evets Uodeed selectios: Selectios that ae made whe the ode of aagemets is ot impotat o elevat
2 Chapte 13 Pemutatios ad Combiatios 717 INTRODUCTION THIS CHAPTER IS AN itoductio to some of the cocepts you will meet i pobability i the HSC Couse. Pobability is the study of how likely it is that somethig will happe. It is used to make pedictios i diffeet aeas, agig fom games of chace to busiess decisiomakig. I this chapte you will study geeal coutig techiques based o the fudametal coutig piciple. These will lead o to the study of pemutatios ad combiatios. These have applicatios i fidig the size of the sample space, o the ways that cetai evets ca happe i pobability. It ca also tell us the umbe of postcodes a state ca have, the umbe of telephoe umbes that is possible i a city ad the umbe of combiatios makig up seial umbes fo appliaces. Fudametal Coutig Piciple Simple pobability You have studied pobability i ealie stages of mathematics. We ca measue pobability i theoy. Howeve, pobability oly gives us a appoximate idea of the likelihood of cetai evets happeig. Fo example, i Lotto daws, thee is a machie that daws out the balls at adom ad a pael of supevisos checks that this happes popely. Each ball is idepedet of the othes ad is equally likely to be daw out. I a hose ace, it is difficult to measue pobability as the hoses ae ot all equally likely to wi. Othe factos such as ability, taiig, expeiece ad weight of the jockey all affect it. The likelihood of ay oe hose wiig is ot adom. The pobability of a evet E happeig, P ( E ), is give by the umbe of ways the evet ca occu, ( E ), compaed with the total umbe of outcomes possible ( S ) (the size of the sample space). E ( ) P ] Eg S () If PE ] g 0 the evet is impossible. If PE ] g 1 the evet is cetai (it has to happe). 0 # PE ] g # 1
3 718 Maths I Focus Mathematics Extesio 1 Pelimiay Couse The sum of all pobabilities is 1. Complemetay evets: P] ot Eg 1 P] Eg o P^L Eh 1  P( E) whee L E is the complemet of E PE ] g + PE ^L h 1 EXAMPLES 1. Aliso buys 5 affle tickets ad 100 ae sold altogethe. What is the pobability that Aliso (a) wis (b) does t wi fist pize i the affle? (a) The size of the sample space, o total umbe of outcomes is 100, sice thee ae 100 tickets altogethe. Aliso has 5 tickets so has 5 diffeet ways of wiig the affle. 5 P ( Wi ) (b) Thee ae o 95 othe tickets that could wi if Aliso loses. 95 P ( Loss ) O, if we kow that the sum of all pobabilities is 1, we could say P( Loss) 1  P( Wi) Thee ae 56 books o music at the school libay ad thee ae 2000 books altogethe. If Athoy selects a book at adom, fid the pobability that it will be a book o music. The size of the sample space is 2000 ad thee ae 56 ways that Athoy could select a music book. 56 P ( Music book )
4 Chapte 13 Pemutatios ad Combiatios Execises 1. A lottey is held i which tickets ae sold. If I buy 2 tickets, what is the pobability of my wiig the pize i the lottey? 2. The pobability of a bus aivig 17 o time is estimated at. What 33 is the pobability that the bus will ot aive o time? 3. The pobability of a seed poducig a pik flowe is 9 7. Fid the pobability that the seed will poduce a diffeet coloued flowe. 7. A shoe shop odes i 20 pais of black, 14 pais of avy ad 3 pais of bow school shoes. If the boxes ae all mixed up, fid the pobability that oe box selected at adom will cotai bow shoes. 8. A biased coi is weighted so that heads comes up twice as ofte as tails. Fid the pobability of tossig a tail. 9. A die has the cete dot paited white o the 5 so that it appeas as a 4. Fid the pobability of ollig (a) a 2 (b) a 4 (c) a umbe less tha I a lottey, tickets ae sold. If Lucia buys 10 tickets, what is the pobability of he wiig fist pize? 5. A machie has a 1.5% chace of beakig dow at ay give time. What is the pobability of the machie ot beakig dow? 6. A bag cotais 6 ed balls ad 8 white balls. If oe ball is daw out at adom, fid the pobability that it will be (a) white (b) ed. 10. A book has 124 pages. If the book is opeed at ay page at adom, fid the pobability of the page umbe beig (a) eithe 80 o 90 (b) a multiple of 10 (c) a odd umbe (d) less tha I the game of pool, thee ae 15 balls, each with the umbe 1 to 15 o it. I Kelly Pool, each peso chooses a umbe at adom fom a cotaie ad has to ty ad sik the ball with the coespodig umbe. If Tacey chooses a umbe, fid the pobability that he ball will be (a) the eight ball (b) a odd umbe (c) a umbe less tha 10. A multiple of 10 is a umbe that is divisible by 10.
5 720 Maths I Focus Mathematics Extesio 1 Pelimiay Couse 12. A box cotaiig a light globe 1 has a pobability of holdig 20 a defective globe. If 160 boxes ae checked, how may globes would be expected to be defective? 13. Thee ae 29 ed, 17 blue, 21 yellow ad 19 gee jelly beas i a packet. If Kate chooses oe at adom, fid the pobability that it will be (a) ed (b) blue o yellow (c) ot gee. 14. The pobability of beedig a 2 white budgie is. If M Seed 15 beeds 240 budgies ove the yea, how may would be expected to be white? 15. A die is olled. Calculate the pobability of ollig (a) a 6 (b) a eve umbe (c) a umbe less tha 3 (d) 4 o moe (e) a multiple of The pobability that a aow will 13 hit a taget is. 18 (a) Fid the pobability that it will miss the taget. (b) If 126 aows ae fied, how may would be expected to hit a taget? 17. A dog ca catch a ball i its mouth 8 times out of 10. (a) What is the pobability of the dog catchig a ball? (b) If the ball is thow to the dog 45 times, how may times would the dog be expected to catch it? 18. I a bag thee ae 21 lollies with pik wappes ad 23 with blue wappes. If Leila chooses a lolly at adom fom the bag, fid the pobability that she selects a lolly with (a) a blue wappe (b) a pik wappe. 19. I a suvey, it was foud that 18 people pefeed Bad A of soft dik while 39 people pefeed Bad B. What is the pobability that a peso chose at adom pefes Bad A? 20. A school has 653 juio ad 329 seio studets. If a studet is chose at adom, what is the pobability that it will be a seio studet? 21. A class has 12 gils ad 19 boys. Eight of the gils ad 12 of the boys play a spot. If a studet is chose at adom, fid the pobability that the peso chose (a) is a boy (b) plays a spot (c) is a gil who does t play a spot (d) does t play a spot. 22. Amie s CD collectio icludes 21 R&B, 14 ock ad 24 jazz albums. If she selects oe CD to play at adom, fid the pobability that the CD is (a) ock (b) R&B o ock (c) jazz (d) ot R&B.
6 Chapte 13 Pemutatios ad Combiatios The pobability of wiig a game of chace is 8 3 ad the 5 pobability of losig is. What 12 is the pobability of a daw? 24. I a poll, 39% said they would vote Labo, 34% said they would vote Libeal ad 8% said they would vote fo idepedets o small paties. What pecetage would vote fo Gees? 25. A aow has a 0.37 pobability of hittig a taget outside the bullseye zoe ad 0.12 pobability of missig the taget altogethe. What is the pobability of the aow hittig the bullseye aea? Coutig techiques I the HSC Couse you will lea about multistage evets (evets whee thee ae two o moe evets such as ollig two dice o tossig two cois). The coutig ca become quite difficult, which is why we itoduce coutig techiques i the Pelimiay Couse. Class Discussio Beak up ito pais ad ty these expeimets with oe doig the activity ad oe ecodig the esults. 1. Toss two cois as may times as you ca i a 5miute peiod ad ecod the esults i the table: Result Two heads Oe head ad oe tail Two tails Tally Compae you esults with othes i the class. What do you otice? Is this supisig? 2. Roll two dice as may times as you ca i a 5miute peiod, fid the total of the two uppemost umbes o the dice ad ecod the esults i the table: Total Tally Compae you esults with othes i the class. What do you otice? Is this supisig? Why do t these esults appea to be equally likely?
7 722 Maths I Focus Mathematics Extesio 1 Pelimiay Couse Thee ae may examples of whee coutig techiques ae useful, i pobability ad i aeas such as maufactuig, busiess, biology ad ecoomics. Fo example, i geetics, the umbe of molecules o DNA stads ca be difficult to fid. Ivestigatio 1. To tavel to wok, Cassie eeds to catch a bus ad a tai. She lives ea a bus stop ad thee ae thee diffeet buses she could catch ito tow. Whe she aives i tow, she eeds to catch oe of fou tais to wok. If thee ae thee buses ad fou tais possible fo Cassie to catch, i how may ways is it possible fo he to tavel to wok? Buses Tais Cassie s house A 1 2 B 3 C 4 2. At a estauat, thee ae thee etees, fou mai meals ad two dessets. Evey time Rick eats at the estauat he chooses to eat a diffeet combiatio of couses. How may times would he eed to go to the estauat to cove all possible combiatios? FUNDAMENTAL COUNTING PRINCIPLE If oe evet ca happe i a diffeet ways, a secod evet ca happe i b diffeet ways, a thid i c diffeet ways ad so o, the these successive evets ca happe i abc diffeet ways. EXAMPLES 1. A pesoal idetificatio umbe (PIN) has 4 lettes followed by 3 umbes. How may diffeet PINs of this type ae possible? Thee ae 26 lettes ad 10 umbes ( 0 9 ) possible fo the positios i the PIN.
8 Chapte 13 Pemutatios ad Combiatios 723 Total umbe 26# 26# 26# 26# 10# 10# # So PINs ae possible. 2. A estauat seves 5 diffeet types of etee, 12 mai couses ad 6 dessets. (a) If I ode ay combiatio of etee, mai couse ad desset at adom, how may diffeet combiatios ae possible? (b) If my fied makes 3 guesses at which combiatio I will ode, what is the pobability that she will guess coectly? (a) Total umbe of combiatios 5# 12# (b) P ^coect guess h Hee ae some examples of coutig whe thee is o epetitio o eplacemet. EXAMPLES 1. To wi a tifecta i a ace, a peso has to pick the hoses that come fist, secod ad thid i the ace. If a ace has 9 hoses, how may diffeet combiatios could be a tifecta? Ay of the 9 hoses could come fist. Ay of the emaiig 8 could come secod. Ay of the emaiig 7 hoses could come thid. Total ways 9# 8 # The pobabilities will be diffeet fo whee each hose will come i the ace, but the umbe of possible diffeet tifecta combiatios will be the same. CONTINUED
9 724 Maths I Focus Mathematics Extesio 1 Pelimiay Couse 2. A goup of 15 people atted a cocet ad 3 of them ae adomly give a fee backstage pass. The fist peso eceives a gold pass, the secod oe a silve pass ad the thid oe a boze pass. I how may diffeet ways ca the passes be give out? Ay of the 15 people ca eceive the fist pass. Thee ae 14 people left who could eceive the secod pass. Similaly thee ae 13 people that could eceive the thid pass. Total umbe of possibilities 15# 14# I Lotto, a machie cotais 45 balls, each with a diffeet umbe fom 1 to 45. (a) I how may ways ca 6 balls be adomly daw? (b) To wi fist pize i Lotto, a peso must choose all 6 umbes coectly. Lisa has 3 tickets i the same daw of Lotto. What is the pobability that she will wi fist pize? (a) The fist ball could be ay of the 45 balls. The secod could be ay of the emaiig 44 balls ad so o. The umbe of ways 45# 44# 43# 42# 41# (b) P ^fist pize h Execises 1. A passwod has 4 lettes. How may combiatios ae possible? 2. A motocycle umbeplate is made up of 2 lettes followed by 2 umbes. How may umbeplates of this type ae available? 3. A passwod ca have up to 5 lettes followed by 4 umbes o it. If I could use ay lette of the alphabet o umbe, how may diffeet passwods could be fomed? Leave you aswe i idex fom.
10 Chapte 13 Pemutatios ad Combiatios A witess saw most of the umbeplate o a getaway ca except fo the fist lette ad the last umbe. How may diffeet cas do the police eed to check i ode to fid this ca? 5. A cetai bad of compute has a seial umbe made up of 10 lettes the 15 umbes. How may computes with this type of seial umbe ca be made? Leave you aswe i idex fom. 6. Victoia has postcodes statig with 3. How may diffeet postcodes ae available i Victoia? 7. A couty tow has telephoe umbes statig with 63 followed by ay 6 othe umbes fom 0 to 9. How may telephoe umbes ae possible i this tow? 8. Yasmi has 12 tops, 5 pais of jeas ad 5 pais of shoes i he wadobe. If she adomly chooses a top, pai of jeas ad shoes, how may combiatios ae possible? 9. A ca maufactue poduces cas i 8 diffeet colous, with eithe maual o automatic gea tasmissio, ad 4 diffeet types of wheels. How may diffeet combiatios ca it poduce? 10. A PIN has 4 umbes. If I foget my PIN I am allowed 3 ties to get it ight. Fid the pobability that I get it withi the 3 ties. 11. A estauat offes 7 mai couses ad 4 dessets, as well as 3 diffeet types of coffee. (a) How may diffeet combiatios of mai couse, desset ad coffee ae possible? (b) Fid the pobability that I adomly pick the combiatio voted most favouite. 12. A telephoe umbe i a capital city ca stat with a 9 ad has 8 digits altogethe. (a) How may telephoe umbes ae possible? (b) If I foget the last 3 digits of my fied s telephoe umbe, how may umbes would I have to ty fo the coect umbe? 13. A compay maufactues compute chips. If it uses a seial umbe o each oe cosistig of 10 lettes, will thee be eough combiatios fo all these chips? 14. A passwod cosists of 2 lettes followed by 5 umbes. What is the pobability that I adomly guess the coect passwod? 15. A city has a populatio of How may digits should its telephoe umbes have so that evey peso ca have oe? 16. A maufactue of compute pats puts a seial umbe o each pat, cosistig of 3 lettes, 4 umbes the 4 lettes. The umbe of pats sold is estimated as 5 millio. Will thee be eough combiatios o this seial umbe to cope with these sales? 17. A bidal shop caies 12 diffeet types of bidal desses, 18 types of veils ad 24 diffeet types of shoes. If Kate chooses a combiatio of dess, veil ad
11 726 Maths I Focus Mathematics Extesio 1 Pelimiay Couse shoes at adom, what is the pobability that she chooses the same combiatio as he fied Jae? 18. Kate chooses a diffeet coloued dess fo each of he 3 bidesmaids. If the colous ae adomly give to each bidesmaid, how may diffeet possibilities ae thee? 3 fee haicuts adomly give away, i how may ways could this be doe? 23. A flock of 28 pelicas is fed 6 fish cacasses. If each cacass is give to a diffeet pelica, i how may ways ca this happe? 19. I a compute ca ace game, the cas that come fist, secod ad thid ae adomly awaded. If thee ae 20 cas, how may possible combiatios of fist, secod ad thid ae thee? 20. Jacquie oly has 4 chocolates left ad decides to adomly choose which of he 6 fieds will eceive oe each. How may possible ways ae thee i which ca she give the chocolates away? 21. Thee pizes ae give away at a cocet by tapig them udeeath adom seats. If thee ae 200 people i the audiece, i how may ways ca these pizes be wo? 22. Thee ae 7 cliets at a haidessig salo. If thee ae 24. A set of cads is umbeed 1 to 100 ad 2 chose at adom. (a) How may diffeet aagemets of odeed pais ae possible? (b) What is the pobability that a paticula odeed pai is chose? 25. Each of 10 cads has a lette witte o it fom A to J. If 3 cads ae selected i ode at adom, fid the pobability that they spell out CAB. Factoial otatio Coutig outcomes whe epetitio o eplacemet is allowed is quite staightfowad, eve whe the umbes become vey lage. EXAMPLE A cad is daw adomly fom a set of 25 cads umbeed 1 to 25 i tu ad the eplaced befoe the ext is selected. How may possible outcomes ae thee if 25 cads ae chose this way? Aswe i scietific otatio, coect to 3 sigificat figues.
12 Chapte 13 Pemutatios ad Combiatios 727 Each time thee is a cad daw, thee ae 25 possibilities. Total umbe 25# 25# 25#...# 25 ] 25 timesg # Whe thee is o epetitio o eplacemet, the calculatios ca be quite log. EXAMPLE A cad is daw adomly fom a set of 25 cads umbeed 1 to 25 i tu without eplacig it befoe the ext is daw. How may possible outcomes ae thee if all 25 cads ae daw out? Aswe i scietific otatio, coect to 3 sigificat figues. Fist cad: thee ae 25 possibilities. Secod cad: thee ae oly 24 possibilities sice oe cad has aleady bee daw out. Thid cad: thee ae 23 possibilities ad so o. Total umbe 25# 24# 23# # 3# 2# # This calculatio is quite tedious! Factoial otatio allows us to easily calculate the umbe of possible outcomes whe selectig all objects i ode with o eplacemet o epetitio.! ]  1g]  2g]  3g]  4g... 3# 2# 1 You ca fid a x! key o most scietific calculatos. Sice the sequece of umbes multiplied does t go futhe tha 1, the by covetio we say that 0! 1 Check 0! o you calculato.
13 728 Maths I Focus Mathematics Extesio 1 Pelimiay Couse EXAMPLES 1. Evaluate (a) 4! (b) 7! (c) 25! (aswe i scietific otatio coect to 3 sigificat figues.) (a) 4! # # # It is much easie to use the x! key o a calculato to fid this. (b) 7! 7 # 6 # 5# 4 # 3# 2 # (c) 25! 1. 55# A goup of 9 teeages is waitig to be seved i a café. They ae each adomly assiged a umbe fom 1 to 9. (a) I how may ways is it possible fo the umbes to be assiged? (b) Oe of the goup eeds to be seved quickly as he has to leave. If he is give the fist umbe, i how may ways is it possible fo the umbes to be assiged? (a) The fist umbe could be assiged 9 ways. The secod umbe could be assiged 8 ways ad so o. Total ways 9! (b) Oe of the goup is give the fist ticket (this ca oly happe i oe way) The secod umbe could be assiged 8 ways ad so o. Total ways 1# 8! Execises 1. Evaluate (a) 6! (b) 10! (c) 0! (d) 8!  7! (e) 5# 4! 7! (f) 4! 12! (g) 5! 13! (h) 4!9! 8! (i) 3!5! 11! (j) 4!7!
14 Chapte 13 Pemutatios ad Combiatios A goup of 9 jockeys ae each give a set of idig colous to wea. If these ae give out i ode adomly, how may diffeet aagemets ae possible? 8. A goup of 7 people lie up to do kaaoke. If they ae each adomly give a sog to sig, how may possible outcomes ae thee? 9. A kidegate class has a abbit, a mouse ad a paot. Thee childe ae selected to take these pets home fo the holidays. If the pets ae adomly give out to these childe, how may diffeet ways ae possible? 10. A PIN cosists of 6 umbes, with o epetitio of ay umbes allowed. How may diffeet PINs ae possible? 3. Each of 6 people at a estauat is give a diffeet coloued glass. How may possible combiatios ae thee? 4. A moutai tail oly has oom fo oe peso at a time. If 12 people ae waitig at the bottom of the tail ad ae adomly picked to stat out, i how may ways ca this happe? 5. A dog walke has 5 dogs ad 5 leashes. I how may diffeet ways is it possible to put a leash o each dog? 6. Thee ae 11 people i a choi ad each eceives a musical scoe. (a) I how may diffeet ways could the scoes be haded out? (b) Russell, the musical diecto, also eeds a musical scoe. I how may ways could the scoes be haded out fo the choi ad the musical diecto? 7. A ow of seats i a theate seats 8 people. I how may ways could a goup of 8 fieds be adomly seated i this ow? 11. I a chous fo a school musical, 7 studets each wea a diffeet mask. I how may diffeet ways ca the masks be wo by these studets? 12. If 15 people play a game of Kelly pool, each peso i tu chooses a umbe at adom betwee 1 ad 15. I how may diffeet ways ca this occu? Aswe i scietific otatio, coect to oe decimal place. 13. (a) A school talet quest has 11 pefomes ad each oe is adomly give the ode i which to pefom. I how may ways ca the ode of pefomaces be aaged? (b) If oe pefome is chose to pefom fist, i how may ways ca the othes be aaged? 14. A goup of 6 fieds sit i the same ow at a cocet. (a) I how may diffeet ways ca they aage themselves? (b) If oe fied must sit o the aisle, i how may ways ca they be aaged?
15 730 Maths I Focus Mathematics Extesio 1 Pelimiay Couse 15. A goup of 8 fieds go to a estauat ad sit at a oud table. If the fist peso ca sit aywhee, i how may ways ca the othes be aaged aoud the table? 16. I a pack of cads, the 4 aces ae take out ad shuffled. (a) What is the pobability of pickig out the Ace of Heats at adom? (b) If all the aces ae aaged i ode, what is the pobability of guessig the coect ode? 17. At a weddig, each of the 12 tables is to have a cetepiece with a diffeet coloued ose. (a) I how may diffeet ways ca the oses be aaged at adom? (b) What is the pobability that the bide will have a pik ose at he table? 18. I a maths exam, a studet has to aage 5 decimals i the coect ode. If he has o idea how to do this ad aages them adomly, what is the pobability that he makes the ight guess fo all the decimals? 19. I a ca ace, the fastest ca is give pole positio ad the othe cas ae adomly give thei statig positios. If thee ae 14 cas altogethe, i how may ways ca this be aaged? 20. Show that 8! (a) 8# 7# 6# 5 4! (b) (c) 11! 6!!! 11# 10# 9# 8# 7 ]  1g]  2g]  3g...] + 1g whee 2! (d) (  )! ]  1g]  2g]  3g...]  + 1g whee 2 Pemutatios Factoial otatio is useful fo fidig the umbe of possible outcomes whe aagig all objects i ode without eplacemet. Howeve, sometimes we eed to fid the umbe of possible outcomes whe aagig oly some of the objects i ode without eplacemet. It is easy to aage objects with eplacemet.
16 Chapte 13 Pemutatios ad Combiatios 731 EXAMPLE I how may ways ca 5 cads be selected fom a total of 20 cads if each oe is eplaced befoe selectig the ext oe? Each selectio ca be made i 20 possible ways. Total 20# 20# 20# 20# ways. Fo selectios fom objects (with epetitios), the umbe of possible outcomes is # # # f ( times) o Howeve, whe aagig objects fom objects i ode without eplacemet, it is ot so easy. EXAMPLE I how may ways ca 5 cads be selected fom a total of 20 cads if thee is o eplacemet? The 1 st cad ca be selected i 20 diffeet ways. The 2 d cad ca be selected i 19 diffeet ways as the fist cad is o loge beig used. The 3 d cad ca be selected i 18 diffeet ways, ad so o. Total 20# 19# 18# 17# ways The calculatios ca become tedious if we select a lage umbe of objects.
17 732 Maths I Focus Mathematics Extesio 1 Pelimiay Couse EXAMPLE If thee ae 20 cads ad 13 cads ae chose i ode at adom without eplacemet, fid the possible umbe of ways the cads ca be chose i scietific otatio coect to 1 decimal place. The fist cad ca be ay of the 20 umbes. The secod cad ca be ay of the emaiig 19 umbes. The thid ca be ay of the emaiig 18 umbes. The umbe of ways the cads ca be chose 20# 19# 18# 17# f# # 10 Fo odeed selectios fom objects without eplacemet, the umbe of possible outcomes is # ] 1g#] 2g#] 3g f ] timesg o ] 1g]  2g]  3g f ]  + 1g A pemutatio descibes a aagemet of objects fom a total of objects i a cetai ode without eplacemet o epetitio. You ca fid a P key o most scietific calculatos. Pemutatio P is the umbe of ways of makig odeed selectios of objects fom a total of objects.! P ]  g! Poof P ] 1g]  2g]  3g f ]  + 1g ]  g]  1g]   2gf 3 $ 2 $ 1 ] 1g]  2g]  3g f ]  + 1g # ]  g]  1g]   2gf 3 $ 2 $ 1 ] 1g]  2g]  3g f]  + 1g]  g]  1g]   2gf3 $ 2 $ 1 ]  g]  1g]   2gf 3 $ 2 $ 1! ]  g! A special case of this esult is: P!
18 Chapte 13 Pemutatios ad Combiatios 733 Poof `! P ]  g!! P ]  g!! 0!! 1! EXAMPLES 1. Evaluate 9 P4 You ca evaluate this o a calculato. 9 9! P 4 ] 94g! 9! 5! 9 $ 8 $ 7 $ 6 $ 5 $ 4 $ 3 $ 2 $ 1 5 $ 4 $ 3 $ 2 $ 1 9 $ 8 $ 7 $ (a) Fid the umbe of aagemets of 3 digits that ca be fomed usig the digits 0 to 9 if each digit ca oly be used oce. (b) How may 3 digit umbes geate tha 700 ca be fomed? (a) Thee ae 10 digits fom 0 to 9. The 1 st digit ca be ay of the 10 digits. The 2 d digit ca be ay of the emaiig 9 digits. The 3 d digit ca be ay of the emaiig 8 digits. Total pemutatios 10# 9# ! o 10 P 3 ] 103g! 10! 7! 720 CONTINUED
19 734 Maths I Focus Mathematics Extesio 1 Pelimiay Couse (b) The 1 st digit must be 7 o 8 o 9 (3 possible digits). The 2 d digit ca be ay of the emaiig 9 digits. The 3 d digit ca be ay of the emaiig 8 digits. Total aagemets 3# 9# Aothe method: Thee ae 3 ways to get the 1 st digit. The possible aagemets of the emaiig 2 digits is 9 P2 9 Total aagemets 3# P2 3# Thee ae some special examples that eed vey caeful coutig, such as aagemets aoud a cicle. Othes ivolve coutig whe thee ae idetical objects. EXAMPLES 1. (a) I how may ways ca 6 people sit aoud a cicula table? (b) If seatig is adom, fid the pobability that 3 paticula people will sit togethe. (a) The 1 st peso ca sit aywhee aoud the table so we oly eed to aage the othe 5 people. The 2 d peso ca sit i ay of the 5 emaiig seats. The 3 d peso ca sit i ay of the emaiig 4 seats ad so o. Total aagemets 5! 120
20 Chapte 13 Pemutatios ad Combiatios 735 (b) The 3 people ca sit aywhee aoud the table togethe i 3# 2# 1 o 3! ways. The emaiig 3 people ca sit togethe i 3! ways. Total aagemets 3! # 3! P ( 3 sit togethe ) I how may ways ca the lettes of the wod EXCEPTIONAL be aaged? EXCEPTIONAL has 11 lettes with the lette E epeated. If each E was diffeet, i.e. E 1 ad E 2, the thee would be 11! aagemets. Howeve, we caot tell the diffeece betwee the 2 Es. Sice thee ae 2! ways of aagig the Es, the thee ae 2! aagemets of the wod EXCEPTIONAL that ae idetical. We eed to divide by 2! to elimiate these idetical aagemets. 11! Total aagemets 2! The umbe of diffeet ways of aagig objects i which a of the objects ae of oe kid, b objects ae of aothe! kid, c of aothe kid ad so o, is give by abc!!! f whee a + b + c + f # EXAMPLE Fid the umbe of ways that the wod ANAETHEMA ca be aaged. Thee ae 9 lettes, icludig 3 As ad 2 Es. Thee ae 9! ways of aagig the lettes, with 3! ways of aagig the As ad 2! ways of aagig the Es. 9! Total aagemets 32!!
21 736 Maths I Focus Mathematics Extesio 1 Pelimiay Couse Some questios ivolvig coutig eed diffeet appoaches ad sometimes it is just a matte of logically wokig it out. EXAMPLES A bag cotais 5 balls of diffeet colous ed, yellow, blue, gee ad white. I how may ways ca these 5 balls be aaged (a) with o estictios (b) if the yellow ball must be fist (c) if the fist ball must ot be ed o white (d) if blue ad gee must be togethe (e) if ed, blue ad gee must be togethe? (a) The 1 st ca be ay of the 5 balls. The 2 d ca be ay of the emaiig 4 balls ad so o. Total aagemets 5! 120 (b) The 1 st ball must be yellow, so thee is oly 1 way of aagig this. The 2 d ball ca be ay of the emaiig 4 balls. The 3 d ball ca be ay of the emaiig 3 balls ad so o. Total aagemets 4! 24 (c) The 1st ball could be yellow, blue o gee so thee ae 3 possible aagemets. The 2 d ball could be ay of the emaiig 4 balls ad so o. Total aagemets 3# 4! 72 (d) Whe two objects must be togethe, we teat them as a sigle object with 2! possible aagemets. So we aage 4 balls i 4! ways: R, Y, BG ad W. But thee ae 2! ways i which to aage the blue ad gee balls. Total aagemets 4! # 2! 48 (e) Whe thee objects ae togethe, we teat them as a sigle object with 3! possible aagemets. We ae the aagig 3 balls i 3! ways: RBG, Y, W. But thee ae 3! ways i which to aage the ed, blue ad gee balls. Total aagemets 3! # 3! 36
22 Chapte 13 Pemutatios ad Combiatios Execises 1. Wite each pemutatio i factoial otatio ad the evaluate. (a) P 6 3 (b) 5 P2 (c) 8 P3 (d) 10 P7 (e) 9 P6 (f) 7 P5 (g) 8 P6 (h) (i) (j) 11 P8 9 P1 6 P6 2. A set of 26 cads, each with a diffeet lette of the alphabet, is placed ito a hat ad cads daw out at adom. Fid the umbe of wods possible if selectig (a) 2 cads (b) 3 cads (c) 4 cads (d) 5 cads. 3. A 3 digit umbe is adomly made fom cads cotaiig the umbes 0 to 9. (a) I how may ways ca this be doe if the cads caot be used moe tha oce ad zeo caot be the fist umbe? (b) How may umbes ove 400 ca be made? (c) How may umbes less tha 300 ca be made? 4. A set of 5 cads, each with a umbe fom 1 to 5 o it, is placed i a box ad 2 daw out at adom. Fid the possible umbe of combiatios (a) altogethe (b) of umbes geate tha 50 possible (c) of odd umbes (d) of eve umbes. 5. (a) How may aagemets of the lettes A, B, C ad D ae possible if o lette ca be used twice? (b) How may aagemets of ay 3 of these lettes ae possible? 6. A 4 digit umbe is to be selected at adom fom the umbes 0 to 9 with o epetitio of digits. (a) How may aagemets ca thee be? (b) How may aagemets of umbes ove 6000 ae thee? (c) How may aagemets of umbes less tha 8000 ae thee? 7. The umbes 1, 2, 3, 4 ad 5 ae aaged i a lie. How may aagemets ae possible if (a) thee is o estictio (b) the umbe is less tha (c) the umbe is geate tha (d) the umbe is odd (e) ay 3 umbes ae selected at adom? 8. Thee ae 12 swimmes i a ace. (a) I how may ways could they fiish? (b) I how may ways could they come i fist, secod ad thid? 9. How may diffeet odeed aagemets ca be made fom the wod COMPUTER with (a) 2 lettes (b) 3 lettes (c) 4 lettes? The fist umbe caot be zeo. The fist umbe is ot zeo.
23 738 Maths I Focus Mathematics Extesio 1 Pelimiay Couse All beads ae diffeet fom each othe. 10. How may diffeet odeed aagemets ca be made fom these wods? (a) CENTIPEDE (b) ALGEBRA (c) TELEVISION (d) ANTARCTICA (e) DONOR (f) BASKETBALL (g) GREEDY (h) DUTIFUL (i) MANUFACTURER (j) AEROPLANE 11. A goup of fieds queue outside a estauat i a staight lie. Fid how may ways the fieds ca be aaged if thee ae (a) 4 fieds (b) 7 fieds (c) 8 fieds (d) 10 fieds (e) 11 fieds. 12. A goup of fieds go ito a estauat ad ae seated aoud a cicula table. Fid how may aagemets ae possible if thee ae (a) 4 fieds (b) 7 fieds (c) 8 fieds (d) 10 fieds (e) 11 fieds. 13. A stig of beads looks the same if tued ove. Fid the umbe of diffeet aagemets possible with (a) 10 beads (b) 12 beads (c) 9 beads (d) 11 beads (e) 13 beads. 14. I how may ways ca a goup of 6 people be aaged (a) i a lie (b) i a cicle? 15. Fid how may diffeet ways a goup of 9 people ca be aaged i (a) a lie (b) a cicle. 16. I how may ways ca a set of 10 beads be aaged (a) i a lie (b) i a cicle aoud the edge of a poste (c) o a bacelet? 17. (a) How may diffeet aagemets ca be made fom the playig cads Jack, Quee, Kig ad Ace? (b) If I choose 2 of these cads adomly, how may diffeet aagemets could I make? (c) If I choose 3 of these cads adomly, how may diffeet aagemets could I make? 18. A goup of 7 people sit aoud a table. I how may ways ca they be aaged (a) with o estictios (b) if 2 people wat to sit togethe (c) if 2 people caot sit togethe (d) if 3 people sit togethe? 19. A goup of 5 boys ad 5 gils lie up outside a ciema. I how may ways ca they be aaged (a) with o estictio? (b) If a paticula gil stads i lie fist? (c) If they alteate betwee boys ad gils (with eithe a gil o boy i fist place)? 20. Fid the pobability that if 10 people sit aoud a table, 2 paticula people will be seated togethe at adom.
24 Chapte 13 Pemutatios ad Combiatios A bookshelf is to hold 5 mathematics books, 8 ovels ad 7 cookbooks. (a) I how may diffeet ways could they be aaged? (Leave aswe i factoial otatio.) (b) If the books ae gouped i categoies, i how may ways ca they be aaged? (Aswe i factoial otatio.) (c) If oe book is chose at adom, fid the pobability that it is a cookbook. 22. (a) How may diffeet aagemets ca be made fom the umbes 3, 4, 4, 5 ad 6? (b) How may aagemets fom umbes geate tha 4000? (c) How may fom umbes less tha 5000? (d) If a aagemet is made at adom, fid the pobability that it is less tha Fid the pobability that a aagemet of the wod LAPTOP will stat with T. 24. What is the pobability that if a 3 lette wod is fomed adomly fom the lettes of PHYSICAL, it will be CAL? 25. A mibus has 6 fowad facig ad 2 backwad facig seats. If 8 people use the bus, i how may ways ca they be seated (a) with o estictios (b) if oe peso must sit i a fowad facig seat (c) if 2 people must sit i a fowad facig seat? 26. If 3 lettes of the wod VALUED ae selected at adom, fid the umbe of possible aagemets if (a) the fist lette is D. (b) the fist lette is a vowel. 27. The lettes of the wod THEORY ae aaged adomly. Fid the umbe of aagemets. (a) with o estictios. (b) if the E is at the begiig. (c) if the fist lette is a cosoat ad the last lette is a vowel. 28. Fid the umbe of aagemets possible if x people ae (a) i a staight lie (b) i a cicle (c) i a cicle with 2 people togethe (d) i a staight lie with 3 people togethe (e) i a cicle with 2 people ot togethe. 29. (a) Use factoial otatio to 8 8 P P 3 5 show that 3! 5! P P  (b) Pove that! ]  g! Pove that P P + P  1 Vowels ae lettes a, e, i, o ad u while cosoats ae all othe lettes.
25 740 Maths I Focus Mathematics Extesio 1 Pelimiay Couse Combiatios The pemutatio P is the umbe of aagemets possible fo a odeed selectio of objects fom a total of objects. Whe the ode is ot impotat, fo example whe AB is the same as BA, the umbe of aagemets is called a combiatio. EXAMPLES 1. A committee of 2 is chose fom Scott, Rachel ad Kate. I how may ways ca this be doe? 3 Numbe of odeed aagemets P2 6 Howeve, a committee of Scott ad Rachel is the same as a committee of Rachel ad Scott. This is the same fo all othe aagemets of the committee. Thee ae 2! ways of aagig each committee of two people. To get the umbe of uodeed aagemets, we divide the umbe of odeed aagemets by 2! 3 P2 Total aagemets 2! 3 2. Thee ae 3 vacacies o a school coucil ad 8 people who ae available. If the vacacies ae filled adomly, i how may ways ca this happe? 8 Numbe of odeedaagemets P3 Howeve, ode is ot ecessay hee, sice the 3 vacacies filled by, say, Hamish, Amie ad Macus, would be the same i ay ode. Thee ae 3! diffeet ways of aagig Hamish, Amie ad Macus. 8 P3 So total aagemets 3! 56 The umbe of ways of makig uodeed selectios of P! objects fom is which is the same as! ]  g!!
26 Chapte 13 Pemutatios ad Combiatios 741 Poof P is the odeed selectio of objects fom objects. Thee ae! ways of aagig objects. If ode is uimpotat, the uodeed selectio of objects fom is give P by!.! P ]  g!!!! 1 # ]  g!!! ]  g!! Combiatio C o a k is the umbe of ways of makig uodeed selectios of objects fom a total of objects.! C ] ! g! We ca call this choose otatio. EXAMPLES 1. A bag cotais 3 white ad 2 black coutes labelled W 1, W 2, W 3 ad B 1, B 2. If two coutes ae daw out of the bag, i how may ways ca this happe if ode is ot impotat? Possible aagemets (uodeed) ae: WW WW WB BB WW WB WB WB WB WB 1 2 Thee ae 10 diffeet combiatios. Usig combiatios, the umbe of diffeet aagemets of choosig 2 coutes fom 5 is 5 C ! C 2 (52)!2! 5! 3!2! 10 CONTINUED
27 742 Maths I Focus Mathematics Extesio 1 Pelimiay Couse 2. If 12 cois ae tossed, fid the umbe of ways of tossig 7 tails. The ode is ot impotat. 12 Thee ae C ways of tossig 7 tails fom 12 cois ! C 7 (127)!7! 12! 5!7! (a) A committee of 5 people is fomed adomly fom a goup of 15 studets. I how may diffeet ways ca the committee be fomed? (b) If the goup cosists of 9 seio ad 6 juio studets, i how may ways ca the committee be fomed if it is to have 3 seio ad 2 juio studets i it? (a) The ode of the committee is ot impotat. Numbe of aagemets b15l (b) 3 seio studets ca be chose i b9l 3 o 84 ways. 2 juio studets ca be chose i b6 2 l o 15 ways. 9 6 Total umbe of aagemets c m# c m # A team of 6 me ad 5 wome is chose at adom fom a goup of 10 me ad 9 wome. If Kaye ad Geg both hope to be chose i the team, fid the pobability that (a) both will be chose (b) eithe will be chose The umbe of possible teams C # C # (a) Fo Kaye to be chose, the 4 out of the othe 8 wome will be chose i.e. 8 C4 Fo Geg to be chose, 5 out of the othe 9 me will be chose i.e. 9 C5
28 Chapte 13 Pemutatios ad Combiatios 743 Numbe of combiatios Pobability 8 9 C # C # (b) Fo Kaye ad Geg ot to be icluded, the 5 out of the othe 8 wome ad 6 out of the othe 9 me will be chose. 8 9 Numbe of combiatios C # C # Pobability Execises 1. Wite i factoial otatio ad evaluate. (a) b9l 5 (b) b 12 l 7 (c) b8l 3 (d) (e) 10 C4 11 C5 2. (a) Evaluate (i) (ii) (iii) (iv) 10 C0 7 C b 9 C9 11 (v) c m 11 (b) Hece complete (i) C 0 (ii) l C 3. Fid the umbe of diffeet ways that a committee of 6 people ca be made adomly fom a goup of (a) 8 people (b) 9 people (c) 11 people (d) 15 people (e) 20 people. 4. (a) A set of 3 ed cads ad 3 blue cads ae placed i a box. By amig the ed cads R 1, R 2 ad R 3 ad the blue cads B 1, B 2 ad B 3, list the umbe of diffeet aagemets possible whe 2 cads ae daw out at adom, with ode ot impotat. How may aagemets ae possible? (b) If thee ae 10 ed ad 10 blue cads ad 7 ae daw out at adom, how may diffeet combiatios ae possible? 5. A coi is tossed 20 times. How may diffeet aagemets ae thee fo tossig 5 heads? The cads ae all distict fom each othe.
29 744 Maths I Focus Mathematics Extesio 1 Pelimiay Couse 6. A set of 10 mables ae placed i a bag ad 6 selected at adom. I how may diffeet ways ca this happe? 7. I poke, 5 cads ae dealt fom a pack of 52 playig cads. How may diffeet aagemets ae possible? 8. Thee cads ae daw adomly fom a set of 10 cads with the umbes 0 to 9 o them. How may diffeet aagemets ae possible if ode is (a) impotat (b) uimpotat? 9. A debatig team of 3 is chose fom a class of 14 studets. I how may ways ca the team be selected if ode is (a) impotat (b) uimpotat? 10. A bag cotais 12 diffeet lollies with blue wappes ad 15 diffeet lollies with ed wappes. If I take 6 lollies out of the bag, how may diffeet combiatios ae possible? 11. A team of 4 playes is chose at adom fom a goup of 20 teis playes to play a exhibitio match. I how may ways could the team be chose? 12. A goup of 3 studets to go o a studet epesetative coucil is chose at adom fom a class of 27. I how may diffeet ways could this be doe? 13. A boad of 8 people is chose fom a membeship of 35. How may diffeet combiatios ae possible? 14. A basketball team of 5 playes is selected at adom fom a goup of 12 PE studets. (a) I how may ways ca the team be selected? (b) Fid the pobability that Eik is selected as oe of the team membes. (c) Fid the pobability that Eik ad Jes ae both selected. 15. A committee of 6 people is to be selected adomly fom a goup of 11 me ad 12 wome. Fid the umbe of possible committees if (a) thee is o estictio o who is o the committee (b) all committee membes ae to be male (c) all membes ae to be female (d) thee ae to be 3 me ad 3 wome (e) a paticula woma is icluded (f) a paticula ma is ot icluded (g) thee ae to be 4 wome ad 2 me. 16. A hose ace has 15 hoses competig ad at the TAB, a quiella pays out o the hoses that come i fist ad secod, i eithe ode. Rya decides to bet o all possible combiatios of quiellas. If it costs him $1 a bet, how much does he pay? 17. A goup of 25 studets cosist of 11 who play a musical istumet ad 14 who do t. Fid the umbe of diffeet aagemets possible if a goup of 9 studets is selected at adom (a) with o estictio (b) who all play musical istumets (c) whee 5 play musical istumets (d) whee 2 do t play musical istumets.
30 Chapte 13 Pemutatios ad Combiatios A set of cads cosists of 8 yellow ad 7 ed cads. (a) If 10 cads ae selected at adom, fid the umbe of diffeet aagemets possible. (b) If 8 cads ae selected, fid the umbe of aagemets of selectig (i) 4 yellow cads (ii) 6 yellow cads (iii) 7 yellow cads (iv) 5 ed cads. 19. Te cads ae selected adomly fom a set of 52 playig cads. Fid the umbe of combiatios selected if (a) thee ae o estictios (aswe i scietific otatio coect to 3 sigificat figues) (b) they ae all heats (c) thee ae 7 heats (d) they ae all ed cads (e) thee ae 4 aces. 20. A aimal efuge has 17 dogs ad 21 cats. If a usig home odes 12 aimals at adom, fid the umbe of ways that the ode would have (a) 7 dogs (b) 9 dogs (c) 10 dogs (d) 4 cats (e) 6 cats. 21. Thee ae 8 white, 9 ed ad 5 blue mables i a bag ad 7 ae daw out at adom. Fid the umbe of aagemets possible (a) with o estictio (b) if all mables ae ed (c) if thee ae 3 white ad 2 ed mables (d) if thee ae 4 ed ad 1 blue mables (e) if thee ae 4 white ad 2 blue mables. 22. Out of a goup of 25 studets, 7 walk to school, 12 catch a tai ad 6 catch a bus. If 6 studets ae selected, fid the umbe of combiatios if (a) all walk to school (b) oe catch a bus (c) 3 walk to school ad 1 catches a bus (d) 1 walks to school ad 4 catch a tai (e) 3 catch a tai ad 1 catches a bus. 23. At a kaaoke ight, a goup of 14 fieds decide that 4 of them will sig a sog togethe. Of the fieds, 5 have peviously sug this sog befoe. I how may ways ca they do this if they select (a) fieds who have all sug the sog peviously (b) 2 of the fieds who sag the sog peviously (c) oe of the fieds who sag the sog peviously? (a) Evaluate C5 12 (b) Evaluate C7 (c) By usig factoial otatio, show why C C 25. By evaluatig both sides, show that C C + C Show that a13k a13k Show that b 10 l b 9 l b l Pove that b l b l Pove that P! C Pove that b l b  1 k  l 1 + b  1 l k k 5 7 The cads ae all distict fom each othe. All mables ae distict.
31 746 Maths I Focus Mathematics Extesio 1 Pelimiay Couse Test youself Fid the umbe of ways of aagig 7 people (a) i a staight lie (b) i a cicle. 2. A bag cotais 8 ed, 5 gee ad 9 yellow mables. If a mable is chose at adom, fid the pobability that it is (a) ed (b) gee o ed (c) ot gee. 3. A caiage has 2 seats facig fowads ad 2 seats facig backwads. Fid the umbe of ways of seatig 4 people i the caiage if (a) thee ae o estictios (b) oe peso must sit facig fowads. 4. A set of 10 cads, umbeed 1 to 10, is placed ito a box ad 3 daw out at adom. Fid the umbe of aagemets possible if ode is (a) impotat (b) uimpotat. 5. A goup of 10 boys ad 16 gils ae o a school excusio. Five of them ae chose at adom to help the teache check the olls. Fid the umbe of ways these studets could be selected if (a) thee is o estictio (b) 3 ae gils (c) all ae boys. 6. The pobability of wiig a game of chace is 59% ad the pobability of a daw is 12%. What is the pobability of losig? 7. I how may ways ca 3 lette wods be selected at adom fom the wod RANDOM? 8. A set of cads umbeed fom 1 to 20 is aaged adomly. I how may ways ca this be doe? (Aswe i scietific otatio, to 2 sigificat figues.) 9. The pobability of a missile hittig a taget is 9 8. What is the pobability of the missile missig the taget? 10. A 5 peso committee is selected fom a class of 30 studets. I how may ways ca the committee be selected? 11. I a hose ace, a peso bets at the TAB o a tifecta. To wi, the peso must pick the fist 3 hoses i ode. I how may ways is it possible to wi if thee ae 11 hoses i the ace? 12. (a) A goup of 9 fieds go out to die ad sit at a oud table. I how may ways ca this be doe? (b) The 9 fieds the go to a ightclub ad adomly queue up to get i. How may aagemets ae thee? (c) If Jack ad Jill queue up togethe, i how may ways ca the fieds lie up? (d) Oce they get ito the ightclub, thee is oly a table fo 3 available, ad the othes will have to stad up. If the fieds adomly assig who sits at the table, i how may diffeet ways ca they be seated? 13. How may diffeet aagemets ae thee of the wod (a) PERMUTATION (b) COMBINATION (c) FACTORIAL (d) PROBABILITY (e) SELECTION
32 Chapte 13 Pemutatios ad Combiatios A set of cois ae tossed. Fid the umbe of ways if tossig k tails. 15. A set of 20 cads is umbeed 1 to 20 ad 6 selected at adom. Fid the umbe of aagemets of selectig (a) all odd umbes (b) the last 2 umbes less tha I Austalia Idyll, thee ae 12 siges who must choose a sog to sig fom a list of 32 sogs. Each sige takes tus i ode to adomly choose a sog. I how may ways could these choices be made? Aswe i scietific otatio coect to 2 decimal places. 17. A ballet class has 30 studets i it. Of these studets, 21 ae pactisig fo a ballet exam. If 8 studets ae chose at adom, fid the umbe of ways that (a) 5 ae pactisig fo the exam (b) all ae pactisig fo the exam (c) 3 ae pactisig fo the exam. 18. Evaluate (a) a6k 4 (b) 9 P7 19. A seial umbe is made up of 4 lettes ad 2 umbes. If zeo is ot allowed, fid how may seial umbes ae possible. 20. (a) Evaluate 0! (b) Show that a k a k 0 Challege Execise Numbes ae fomed fom the digits 1, 2, 3, 3, ad 7 at adom. (a) I how may ways ca they be aaged with o estictios? (b) I how may ways ca they be aaged to fom a umbe geate tha ? 2. A cham bacelet has 6 chams o it. I how may ways ca the chams be aaged if the bacelet (a) has a clasp (b) has o clasp? 3. Show that b l k b l k b l k fo 1# k # A goup of people sit aoud a cicula table. (a) I how may ways ca they be aaged? (b) How may aagemets ae possible if k people sit togethe? 5. (a) How may diffeet aagemets of the wod CHALLENGE ae thee? (b) How may diffeet aagemets ae possible if 3 lettes ae adomly selected fom the wod CHALLENGE ad aaged ito wods? 6. A subcommittee of 5 people is fomed fom the 12 membes of a boad. (a) If this is a adom selectio, i how may diffeet ways ca the committee be fomed? (b) If thee ae 4 NSW membes ad 3 Queeslad membes o the boad, what is the pobability that 2 NSW ad 2 Queeslad membes will be o the committee?
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