# Permutations and Combinations

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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 decisio-makig. 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 )

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.

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!

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.

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 ] 9-4g! 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 ] 10-3g! 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

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 (5-2)!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 (12-7)!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

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