PERMUTATIONS AND COMBINATIONS

Transcription

1 Pemutatios Ad ombiatios MODULE - I 7 PERMUTATIONS AND OMBINATIONS The othe day, I wated to tavel fom Bagaloe to Allahabad by tai. Thee is o diect tai fom Bagaloe to Allahabad, but thee ae tais fom Bagaloe to Itasi ad fom Itasi to Allahabad. Fom the ailway timetable I foud that thee ae two tais fom Bagaloe to Itasi ad thee tais fom Itasi to Allahabad. Now, i how may ways ca I tavel fom Bagaloe to Allahabad? Thee ae coutig poblems which come ude the bach of Mathematics called combiatoics. Suppose you have five jas of spices that you wat to aage o a shelf i you kitche. You would like to aage the jas, say thee of them, that you will be usig ofte i a moe accessible positio ad the emaiig two jas i a less accessible positio. I how may ways ca you do it? I aothe situatio suppose you ae paitig you house. If a paticula shade o colou is ot available, you may be able to ceate it by mixig diffeet colous ad shades. While ceatig ew colous this way, the ode of mixig is ot impotat. It is the combiatio o choice of colous that detemie the ew colous; but ot the ode of mixig. To give aothe simila example, whe you go fo a jouey, you may ot take all you desses with you. You may have sets of shits ad touses, but you may take oly sets. I such a case you ae choosig out of sets ad the ode of choosig the sets does t matte. I these examples, we eed to fid out the umbe of choices i which it ca be doe. I this lesso we shall coside simple coutig methods ad use them i solvig such simple coutig poblems. MATHEMATIS 7

2 MODULE - I OBJETIVES Pemutatios Ad ombiatios Afte studyig this lesso, you will be able to : fid out the umbe of ways i which a give umbe of objects ca be aaged; state the Fudametal Piciple of outig; defie! ad evaluate it fo deffeet values of ; state that pemutatio is a aagemet ad wite the meaig of P ; state that P! ()! ad apply this to solve poblems; 1 show that ()( 1)()() + i + P P ii P P ; + 1 state that a combiatio is a selectio ad wite the meaig of ; distiguish betwee pemutatios ad combiatios; deive!!()! ad apply the esult to solve poblems; deive the elatio P! ; veify that ad give its itepetatio; ad deive ad apply the esult to solve poblems. EXPETED BAKGROUND KNOWLEDGE Numbe Systems Fou Fudametal Opeatios 7.1OUNTING PRINIPLE Let us ow solve the poblem metioed i the itoductio. We will witet 1, t to deote tais fom Bagaloe to Itasi ad T 1, T, T, fo the tais fom Itasi to Allahabad. Suppose I take t 1 to tavel fom Bagaloe to Itasi. The fom Itasi I ca taket 1 o T o T. So the possibilities ae t 1 T 1, t T ad t T whee t 1 T 1 deotes tavel fom Bagaloe to Itasi byt 1 ad tavel fom Itasi to Allahabad by T 1. Similaly, if I take t to tavel fom Bagaloe to Itasi, the the possibilities ae t T 1, t T ad t T. Thus, i all thee ae 6( ) possible ways of tavellig fom Bagaloe to Allahabad. Hee we had a small umbe of tais ad thus could list all possibilities. Had thee bee 10 tais fom Bagaloe to Itasi ad 1 tais fom Itasi to Allahabad, the task would have bee 8 MATHEMATIS

3 Pemutatios Ad ombiatios vey tedious. Hee the Fudametal Piciple of outig o simply the outig Piciple comes i use : If ay evet ca occu i m ways ad afte it happes i ay oe of these ways, a secod evet ca occu i ways, the both the evets togethe ca occu i m ways. Example 7.1 How may multiples of ae thee fom 10 to 9? Solutio : As you kow, multiples of ae iteges havig 0 o i the digit to the exteme ight (i.e. the uit s place). The fist digit fom the ight ca be chose i ways. The secod digit ca be ay oe of 1,,,,,6,7,8,9. i.e. Thee ae 9 choices fo the secod digit. MODULE - I Thus, thee ae 9 18 multiples of fom 10 to 9. Example 7. I a city, the bus oute umbes cosist of a atual umbe less tha 100, followed by oe of the lettes A,B,,D,E ad F. How may diffeet bus outes ae possible? Solutio : The umbe ca be ay oe of the atual umbes fom 1 to 99. Thee ae 99 choices fo the umbe. The lette ca be chose i 6 ways. Numbe of possible bus outes ae HEK YOUR PROGRESS (a) How may digit umbes ae multiples of? (b) A coi is tossed thice. How may possible outcomes ae thee? (c) If you have shits ad pais of touses ad ay shit ca be wo with ay pai of touses, i how may ways ca you wea you shits ad pais of touses? (d) A touist wats to go to aothe couty by ship ad etu by ai. She has a choice of diffeet ships to go by ad ailies to etu by. I how may ways ca she pefom the jouey?. (a) I how may ways ca two vacacies be filled fom amog me ad 1 wome if oe vacacy is filled by a ma ad the othe by a woma? (b) Flooig ad paitig of the walls of a oom eeds to be doe. The flooig ca be doe i colous ad paitig of walls ca be doe i 1 colous. If ay colou combiatio is allowed, fid the umbe of ways of flooig ad paitig the walls of the oom. So fa, we have applied the coutig piciple fo two evets. But it ca be exteded to thee o moe, as you ca see fom the followig examples : MATHEMATIS 9

4 MODULE - I Pemutatios Ad ombiatios Example 7. Thee ae questios i a questio pape. If the questios have, ad solutiosvely, fid the total umbe of solutios. Solutio : Hee questio 1 has solutios, questio has solutios ad questio has solutios. By the multiplicatio (coutig) ule, total umbe of solutios Example 7. oside the wod ROTOR. Whicheve way you ead it, fom left to ight o fom ight to left, you get the same wod. Such a wod is kow as palidome. Fid the maximum possible umbe of -lette palidomes. Solutio : The fist lette fom the ight ca be chose i 6 ways because thee ae 6 alphabets. Havig chose this, the secod lette ca be chose i 6 ways The fist two lettes ca chose i ways Havig chose the fist two lettes, the thid lette ca be chose i 6 ways. All the thee lettes ca be chose i ways. It implies that the maximum possible umbe of five lette palidomes is 1776 because the fouth lette is the same as the secod lette ad the fifth lette is the same as the fist lette. Note : I Example 7. we foud the maximum possible umbe of five lette palidomes. Thee caot be moe tha But this does ot mea that thee ae 1776 palidomes. Because some of the choices like may ot be meaigful wods i the Eglish laguage. Example 7. How may -digit umbes ca be fomed with the digits 1,,7,8 ad 9 if the digits ae ot epeated. Solutio : Thee digit umbe will have uit s, te s ad huded s place. Out of give digits ay oe ca take the uit s place. This ca be doe i ways.... (i) Afte fillig the uit s place, ay of the fou emaiig digits ca take the te s place. This ca be doe i ways.... (ii) Afte fillig i te s place, huded s place ca be filled fom ay of the thee emaiig digits. 0 MATHEMATIS

5 Pemutatios Ad ombiatios This ca be doe i ways.... (iii) By coutig piciple, the umbe of digit umbes 60 Let us ow state the Geeal outig Piciple If thee ae evets ad if the fist evet ca occu i m 1 ways, the secod evet ca occu i m ways afte the fist evet has occued, the thid evet ca occu i m ways afte the secod evet has ocued, ad so o, the all the evets ca occu i m m m m 1 1 ways. MODULE - I Example 7.6 Suppose you ca tavel fom a place A to a place B by buses, fom place B to place by buses, fom place to place D by buses ad fom place D to place E by buses. I how may ways ca you tavel fom A to E? Solutio : The bus fom A to B ca be selected i ways. The bus fom B to ca be selected i ways. The bus fom to D ca be selected i ways. The bus fom D to E ca be selected i ways. So, by the Geeal outig Piciple, oe ca tavel fom A to E i ways 7 ways. HEK YOUR PROGRESS (a) What is the maximum umbe of 6-lette palidomes? (b) What is the umbe of 6-digit palidomic umbes which do ot have 0 i the fist digit?. (a) I a school thee ae Eglish teaches, 7 Hidi teaches ad Fech teaches. A thee membe committee is to be fomed with oe teache epesetig each laguage. I how may ways ca this be doe? (b) I a college studets uio electio, studets ae cotestig fo the post of Pesidet. studets ae cotestig fo the post of Vice-pesidet ad studets ae cotestig fo the post of Secetay. Fid the umbe of possible esults.. (a) How may thee digit umbes geate tha 600 ca be fomed usig the digits 1,,,6,8 without epeatig the digits? (b) A peso wats to make a time table fo peiods. He has to fix oe peiod each fo Eglish, Mathematics, Ecoomics ad ommece. How may diffeet time tables ca he make? 7. PERMUTATIONS Suppose you wat to aage you books o a shelf. If you have oly oe book, thee is oly MATHEMATIS 1

6 MODULE - I Pemutatios Ad ombiatios oe way of aagig it. Suppose you have two books, oe of Histoy ad oe of Geogaphy. You ca aage the Geogaphy ad Histoy books i two ways. Geogaphy book fist ad the Histoy book ext, GH o Histoy book fist ad Geogaphy book ext; HG. I othe wods, thee ae two aagemets of the two books. Now, suppose you wat to add a Mathematics book also to the shelf. Afte aagig Histoy ad Geogaphy books i oe of the two ways, saygh, you ca put Mathematics book i oe of the followig ways: MGH, GMH o GHM. Similaly, coespodig to HG, you have thee othe ways of aagig the books. So, by the outig Piciple, you ca aage Mathematics, Geogaphy ad Histoy books i ways 6 ways. By pemutatio we mea a aagemet of objects i a paticula ode. I the above example, we wee discussig the umbe of pemutatios of oe book o two books. I geeal, if you wat to fid the umbe of pemutatios of objects 1, how ca you do it? Let us see if we ca fid a aswe to this. Simila to what we saw i the case of books, thee is oe pemutatio of 1 object, 1 pemutatios of two objects ad 1 pemutatios of objects. It may be that, thee ae ( 1) ( )... 1 pemutatios of objects. I fact, it is so, as you will see whe we pove the followig esult. Theoem 7.1 The total umbe of pemutatios of objects is ( 1)...1. Poof : We have to fid the umbe of possible aagemets of diffeet objects. The fist place i a aagemet ca be filled i diffeet ways. Oce it has bee doe, the secod place ca be filled by ay of the emaiig ( 1) objects ad so this ca be doe i ( 1) ways. Similaly, oce the fist two places have bee filled, the thid ca be filled i ( ) ways ad so o. The last place i the aagemet ca be filled oly i oe way, because i this case we ae left with oly oe object. Usig the coutig piciple, the total umbe of aagemets of diffeet objects is ( 1)( ) (7.1) The poduct ( 1)....1 occus so ofte i Mathematics that it deseves a ame ad otatio. It is usually deoted by! (o by ead as factoial).! ( 1) Hee is a example to help you familiaise youself with this otatio. Example 7.7 Evaluate (a)! (b)! +! (c)!! Solutio : (a)! 1 6 (b)! 1! 1 MATHEMATIS

7 Pemutatios Ad ombiatios Theefoe,! +! + 6 (c)!! 6 1 MODULE - I Notice that! satisfies the elatio ( )! 1!... (7.) This is because, ( 1)! [( 1). ( )....1]. ( 1). ( )...1! Of couse, the above elatio is valid oly fo because 0! has ot bee defied so fa. Let us see if we ca defie 0! to be cosistet with the elatio. I fact, if we defie 0! 1... (7.) the the elatio 7. holds fo 1 also. Example 7.8 Suppose you wat to aage you Eglish, Hidi, Mathematics, Histoy, Geogaphy ad Sciece books o a shelf. I how may ways ca you do it? Solutio : We have to aage 6 books. The umbe of pemutatios of objects is!. ( 1). ( )....1 Hee 6 ad theefoe, umbe of pemutatios is HEK YOUR PROGRESS (a) Evaluate : (i) 6! (ii) 7! (iii) 7! +! (iv) 6!! (v) (b) Which of the followig statemets ae tue? (i)!! 6! (ii)! +! 6! (iii)! divides! (iv)! -!!. (a) studets ae stayig i a domitoy. I how may ways ca you allot beds to them?!!.! (b) I how may ways ca the lettes of the wod TRIANGLE be aaged? (c) How may fou digit umbes ca be fomed with digits 1,, ad ad with distict digits? 7. PERMUTATION OF OBJETS OUT OF OBJETS Suppose you have five stoy books ad you wat to distibute oe each to Asha, Akhta ad Jasvide. I how may ways ca you do it? You ca give ay oe of the five books to Asha MATHEMATIS

8 MODULE - I Pemutatios Ad ombiatios ad afte that you ca give ay oe of the emaiig fou books to Akhta. Afte that, you ca give oe of the emaiig thee books to Jasvide. So, by the outig Piciple, you ca distibute the books i ie.60 ways. Moe geeally, suppose you have to aage objects out of objects. I how may ways ca you do it? Let us view this i the followig way. Suppose you have objects ad you have to aage of these i boxes, oe object i each box. ways 1 ways ways boxes Fig. 7.1 Suppose thee is oe box. 1. You ca put ay of the objects i it ad this ca be doe i ways. Suppose thee ae two boxes.. You ca put ay of the objects i the fist box ad afte that the secod box ca be filled with ay of the emaiig 1 objects. So, by the coutig piciple, the two boxes ca be filled i ( 1) ways. Similaly, boxes ca be filled i ( 1) ( ) ways. I geeal, we have the followig theoem. Theoem 7. The umbe of pemutatios of objects out of objects is ( 1) ( + 1). The umbe of pemutatios of objects out of objects is usually deoted by Thus, ( 1 )( )...( + 1)... (7.) P P. Poof : Suppose we have to aage objects out of diffeet objects. I fact it is equivalet to fillig places, each with oe of the objects out of the give objects. The fist place ca be filled i diffeet ways. Oce this has bee doe, the secod place ca be filled by ay oe of the emaiig ( 1) objects, i ( 1) ways. Similaly, the thid place ca be filled i ( ) ways ad so o. The last place, theth place ca be filled i [ ( 1)] i.e. ( +1) diffeet ways. You may easily see, as to why this is so. Usig the outig Piciple, we get the equied umbe of aagemets of out of objects is ( 1) ( )...( + 1) Example 7.9 Evaluate : (a) P (b) 6 P (c) P P (d) 6 P P Solutio : (a) P ( 1) 1. 6 (b) P 6(6 1) (6 ) MATHEMATIS

9 Pemutatios Ad ombiatios P ( 1)( ) (c) P ( 1) MODULE - I 6 (d) P P 6(6 1)(6 ) ( 1) 6 00 Example 7.10 If you have 6 New Yea geetig cads ad you wat to sed them to of you fieds, i how may ways ca this be doe? Solutio : We have to fid umbe of pemutatios of objects out of 6 objects. 6 This umbe is P 6(6 1)(6 )(6 ) Theefoe, cads ca be set i 60 ways. oside the fomula fo P, amely, P ( 1)... ( + 1). This ca be obtaied by emovig the tems, 1,...,, 1 fom the poduct fo!. The poduct of these tems is ( ) ( 1)...1, i.e., ( )!. Now,! ( )! ( 1) ( )...( + 1) ( )...1 ( ) ( 1)...1 ( 1) ( )...( + 1) P So, usig the factoial otatio, this fomula ca be witte as follows : P!... (7.) ( )! Example 7.11 Fid the value of P0. Solutio : Hee 0. Usig elatio 7., we get! P0 1! Example 7.1 Show that ( + 1) P +1 P +1 Solutio : ( + 1) P! ( + 1) ( )! ( + 1)! ( )! ( + 1)! [ ]! ( + 1) ( + 1) [ witig as [( + 1) ( + 1)] +1 P +1 (By defiitio) MATHEMATIS

10 MODULE - I HEK YOUR PROGRESS 7. Pemutatios Ad ombiatios 1. (a) Evaluate : (i) P (ii) 6 P (iii) (b) Veify each of the followig statemets : P P (iv) 6 P P (v) P (i) 6 P P (ii) P P 1 (iii) P P P (iii) 7 P + P P. (a) (i) What is the maximum possible umbe of - lette wods i Eglish that do ot cotai ay vowel? (ii) What is the maximum possible umbe of - lette wods i Eglish which do ot have ay vowel othe tha a? (b) Suppose you have cots ad bedspeads i you house. I how may ways ca you put the bedspeads o you cots? (c) You wat to sed Diwali Geetigs to fieds ad you have 7 geetig cads with you. I how may ways ca you do it?. Show that P P 1.. Show that ( ) P P PERMUTATIONS UNDER SOME ONDITIONS We will ow see examples ivolvig pemutatios with some exta coditios. Example 7.1 Suppose 7 studets ae stayig i a hall i a hostel ad they ae allotted 7 beds. Amog them, Pavi does ot wat a bed ext to Aju because Aju soes. The, i how may ways ca you allot the beds? Solutio : Let the beds be umbeed 1 to 7. ase 1 : Suppose Aju is allotted bed umbe 1. The, Pavi caot be allotted bed umbe. So Pavi ca be allotted a bed i ways. Afte allotig a bed to Pavi, the emaiig studets ca be allotted beds i! ways. So, i this case the beds ca be allotted i! ways 600 ways. ase : Aju is allotted bed umbe 7. The, Pavi caot be allotted bed umbe 6 As i ase 1, the beds ca be allotted i 600 ways. 6 MATHEMATIS

11 Pemutatios Ad ombiatios ase : Aju is allotted oe of the beds umbeed,,, o 6. Pavi caot be allotted the beds o the ight had side ad left had side of Aju s bed. Fo example, if Aju is allotted bed umbe, beds umbeed 1 o caot be allotted to Pavi. Theefoe, Pavi ca be allotted a bed i ways i all these cases. Afte allottig a bed to Pavi, the othe ca be allotted a bed i! ways. Theefoe, i each of these cases, the beds ca be allotted i! 80 ways. MODULE - I The beds ca be allotted i ( )(100 ways 00) ways ways. Example 7.1 I how may ways ca a aimal taie aage lios ad tiges i a ow so that o two lios ae togethe? Solutio : They have to be aaged i the followig way : L T L T L T L T L The lios should be aaged i the places maked L. This ca be doe i! ways. The tiges should be i the places maked T. This ca be doe i! ways. Theefoe, the lios ad the tiges ca be aaged i!! ways 880 ways. Example 7.1 Thee ae books o faiy tales, ovels ad plays. I how may ways ca you aage these so that books o faiy tales ae togethe, ovels ae togethe ad plays ae togethe ad i the ode, books o faiytales, ovels ad plays. Solutio : Thee ae books o faiy tales ad they have to be put togethe. They ca be aaged i! ways. Similaly, thee ae ovels. They ca be aaged i! ways. Ad thee ae plays. They ca be aaged i! ways. So, by the coutig piciple all of them togethe ca be aaged i!!! ways 1780 ways. Example 7.16 Suppose thee ae books o faiy tales, ovels ad plays as i Example 7.1. They have to be aaged so that the books o faiy tales ae togethe, ovels ae togethe ad plays ae togethe, but we o loge equie that they should be i a specific ode. I how may ways ca this be doe? MATHEMATIS 7

12 MODULE - I Pemutatios Ad ombiatios Solutio : Fist, we coside the books o faiy tales, ovels ad plays as sigle objects. These thee objects ca be aaged i!ways 6 ways. Let us fix oe of these 6 aagemets. This may give us a specific ode, say, ovels faiy tales plays. Give this ode, the books o the same subject ca be aaged as follows. The books o faiy tales ca be aaged amog themselves i! ways. The ovels ca be aaged i! 10 ways. The plays ca be aaged i! 6 ways. Fo a give ode, the books ca be aaged i ways. Theefoe, fo all the 6 possible odes the books ca be aaged i ways. Example 7.17 I how may ways ca gils ad boys be aaged i a ow so that all the fou gils ae togethe? Solutio : Let gils be oe uit ad ow thee ae 6 uits i all. They ca be aaged i 6! ways. I each of these aagemets gils ca be aaged i! ways. Total umbe of aagemets i which gils ae always togethe 6!! Example 7.18 How may aagemets of the lettes of the wod BENGALI ca be made (i) if the vowels ae eve togethe. (ii) if the vowels ae to occupy oly odd places. Solutio : Thee ae 7 lettes i the wod Begali; of these ae vowels ad cosoats. (i) osideig vowels a, e, i as oe lette, we ca aage +1 lettes i! ways i each of which vowels ae togethe. These vowels ca be aaged amog themselves i! ways. Total umbe of wods!! (ii) Thee ae odd places ad eve places. vowels ca occupy odd places i P ways ad costats ca be aaged i P ways. 8 MATHEMATIS

13 Pemutatios Ad ombiatios Numbe of wods P P MODULE - I 76. HEK YOUR PROGRESS M. Gupta with Ms. Gupta ad thei fou childe is tavellig by tai. Two lowe beths, two middle beths ad uppe beths have bee allotted to them. M. Gupta has udegoe a kee sugey ad eeds a lowe beth while Ms. Gupta wats to est duig the jouey ad eeds a uppe beth. I how may ways ca the beths be shaed by the family?. oside the wod UNBIASED. How may wods ca be fomed with the lettes of the wod i which o two vowels ae togethe?. Thee ae books o Mathematics, books o Eglish ad 6 books o Sciece. I how may ways ca you aage them so that books o the same subject ae togethe ad they ae aaged i the ode Mathematics Eglish Sciece.. Thee ae Physics books, hemisty books, Botay books ad Zoology books. I how may ways ca you aage them so that the books o the same subject ae togethe?. boys ad gils ae to be seated i 7 chais such that o two boys ae togethe. I how may ways ca this be doe? 6. Fid the umbe of pemutatios of the lettes of the wod TENDULKAR, i each of the followig cases : (i) begiig with T ad edig with R. (ii) vowels ae always togethe. (iii) vowels ae eve togethe. 7. OMBINATIONS Let us coside the example of shits ad touses as stated i the itoductio. Thee you have sets of shits ad touses ad you wat to take sets with you while goig o a tip. I how may ways ca you do it? Let us deote the sets bys 1, S, S, S. The you ca choose two pais i the followig ways : 1. { S 1,S }. { S 1, S }. { S 1,S }. { S, S }. { S } 6. { S, S } [Obseve that { } 1,S, S S is the same as { S },S 1 ]. So, thee ae 6 ways of choosig the two sets that you wat to take with you. Of couse, if you had 10 pais ad you wated to take 7 pais, it will be much moe difficult to wok out the umbe of pais i this way. MATHEMATIS 9

14 MODULE - I Pemutatios Ad ombiatios Now as you may wat to kow the umbe of ways of weaig out of sets fo two days, say Moday ad Tuesday, ad the ode of weaig is also impotat to you. We kow fom sectio 7., that it ca be doe i P 1 ways. But ote that each choice of sets gives us two ways of weaig sets out of sets as show below : 1. {S 1,S } S 1 o Moday ad S o Tuesday o S o Moday ad S 1 o Tuesday. {S 1,S } S 1 o Moday ad S o Tuesday o S o Moday ad S 1 o Tuesday. {S 1,S } S 1 o Moday ad S o Tuesday o S o Moday ad S 1 o Tuesday. {S,S } S o Moday ad S o Tuesday o S o Moday ad S o Tuesday. {S,S } S o Moday ad S o Tuesday o S o Moday ad S o Tuesday 6. {S,S } S o Moday ad S o Tuesday o S o Moday ad S o Tuesday Thus, thee ae 1 ways of weaig out of pais. This agumet holds good i geeal as we ca see fom the followig theoem. Theoem 7. Let 1 be a itege ad. Let us deote the umbe of ways of choosig objects out of objects by. The P... (7.6)! Poof : We ca choose objects out of objects i ways. Each of the objects chose ca be aaged i! ways. The umbe of ways of aagig objects is!. Thus, by the coutig piciple, the umbe of ways of choosig objects ad aagig the objects chose ca be doe i! ways. But, this is pecisely P. I othe wods, we have P!.... (7.7) Dividig both sides by!, we get the esult i the theoem. Hee is a example to help you to familiaise youself with Example 7.19 Evaluate each of the followig : (a) (b). (c) + (d) 6. Solutio : (a) P (b) P 10.! 1.! MATHEMATIS

15 Pemutatios Ad ombiatios P P... (c) !! (d) P. 0 ad 6! MODULE - I Example 7.0 Fid the umbe of subsets of the set {1,,,,,6,7,8,9,10,11} havig elemets. Solutio : Hee the ode of choosig the elemets does t matte ad this is a poblem i combiatios. We have to fid the umbe of ways of choosig elemets of this set which has 11 elemets. By elatio (7.6), this ca be doe i ways Example poits lie o a cicle. How may cyclic quadilateals ca be daw by usig these poits? Solutio : Fo ay set of poits we get a cyclic quadilateal. Numbe of ways of choosig 1 poits out of 1 poits is 9. Theefoe, we ca daw 9 quadilateals. Example 7. I a box, thee ae black pes, white pes ad ed pes. I how may ways ca black pes, white pes ad ed pes ca be chose? Solutio : Numbe of ways of choosig black pes fom black pes P. 10.! 1. Numbe of ways of choosig white pes fom white pes P..! 1. Numbe of ways of choosig ed pes fom ed pes P. 6.! 1. MATHEMATIS 1

16 MODULE - I Pemutatios Ad ombiatios By the outig Piciple, black pes, white pes, ad ed pes ca be chose i ways. Example 7. A questio pape cosists of 10 questios divided ito two pats A ad B. Each pat cotais five questios. A cadidate is equied to attempt six questios i all of which at least should be fom pat A ad at least fom pat B. I how may ways ca the cadidate select the questios if he ca aswe all questios equally well? Solutio : The cadidate has to select six questios i all of which at least two should be fom Pat A ad two should be fom Pat B. He ca select questios i ay of the followig ways : Pat A (i ) (ii ) (iii ) Pat B If the cadidate follows choice (i), the umbe of ways i which he ca do so is 10 0 If the cadidate follows choice (ii), the umbe of ways i which he ca do so is Similaly, if the cadidate follows choice (iii), the the umbe of ways i which he ca do so is 0. Theefoe, the cadidate ca select the questio i ways. Example 7. A committee of pesos is to be fomed fom 6 me ad wome. I how may ways ca this be doe whe (i ) at least wome ae icluded? (ii ) atmost wome ae icluded? Solutio : (i) Whe at least wome ae icluded. The committee may cosist of 6 wome, me : It ca be doe i ways. 6 o, wome, 1 ma : It ca be doe i ways. 1 6 o, wome, me : It ca be doe i ways. Total umbe of ways of fomig the committee MATHEMATIS

17 Pemutatios Ad ombiatios (ii ) Whe atmost wome ae icluded The committee may cosist of MODULE - I 6 wome, me : It ca be doe i. ways 6 o, 1 woma, me : It ca be doe i 1. ways o, me : It ca be doe i 6 ways Total umbe of ways of fomig the committee Example 7. The Idia icket team cosists of 16 playes. It icludes wicket keepes ad bowles. I how may ways ca a cicket eleve be selected if we have to select 1 wicket keepe ad atleast bowles? Solutio : We ae to choose 11 playes icludig 1 wicket keepe ad bowles o, 1 wicket keepe ad bowles. Numbe of ways of selectig 1 wicket keepe, bowles ad 6 othe playes Numbe of ways of selectig 1 wicket keepe, bowles ad othe playes Total umbe of ways of selectig the team MATHEMATIS

18 MODULE - I HEK YOUR PROGRESS (a) Evaluate : Pemutatios Ad ombiatios (i) 1 (ii) (iii) + (iv) 9 6 (b) Veify each of the followig statemet : 1 (i) (ii) 6 (iii) (iv) + 8. Fid the umbe of subsets of the set {1,,, 7, 9, 11, 1,..., }each havig elemets.. Thee ae 1 poits lyig o a cicle. How may petagos ca be daw usig these poits?. I a fuit basket thee ae apples, 7 plums ad 11 oages. You have to pick fuits of each type. I how may ways ca you make you choice?. A questio pape cosists of 1 questios divided ito two patsa ad B, cotaiig ad 7 questios epectively. A studet is equied to attempt 6 questios i all, selectig at least fom each pat. I how may ways ca a studet select a questio? 6. Out of me ad wome, a committee of pesos is to be fomed. I how may ways ca it be fomed selectig (i) exactly 1 woma. (ii) atleast 1 woma. 7. A cicket team cosists of 17 playes. It icludes wicket keepes ad bowles. I how may ways ca a playig eleve be selected if we have to select 1 wicket keepe ad atleast bowles? 8. To fill up vacacies, applicatios wee ecieved. Thee wee 7 S.. ad 8 O.B.. cadidates amog the applicats. If posts wee eseved fo S.. ad 1 fo O.B.. cadidates, fid the umbe of ways i which selectio could be made? 7.6 SOME SIMPLE PROPERTIES OF I this sectio we will pove some simple popeties of which will make the computatios of these coefficiets simple. Let us go back agai to Theoem 7.. Usig elatio 7.7 we ca ewite the fomula fo as follows :!...(7.8)!( )! MATHEMATIS

19 Pemutatios Ad ombiatios Example 7.6 Fid the value of 0.! 1 Solutio : Hee 0. Theefoe, 0 1, 0!! 0! sice we have defied 0! 1. The fomula give i Theoem 7. was used i the pevious sectio. As we will see shotly, the fomula give i Equatio 7.8 will be useful fo povig cetai popeties of. MODULE - I...(7.9) This meas just that the umbe of ways of choosig objects out of objects is the same as the umbe of ways of ot choosig( ) objects out of objects. I the example descibed i the itoductio, it just meas that the umbe of ways of selectig sets of desses is the same as the umbe of ways of ejectig desses. I Example 7.0, this meas that the umbe of ways of choosig subsets with elemets is the same as the umbe of ways of ejectig 8 elemets sice choosig a paticula subset of elemets is equivalet to ejectig its complemet, which has 8 elemets. Let us ow pove this elatio usig Equatio 7.8. The deomiato of the ight had side of this equatio is! ( )!. This does ot chage whe we eplace by. ()!.[()]!()!.! The umeato is idepedet of. Theefoe, eplacig by i Equatio 7.8 we get esult. How is the elatio 7.9 useful? Usig this fomula, we get, fo example, is the same as 100. The secod value is much moe easie to calculate tha the fist oe. Example 7.7 Evaluate : (a) 7 (c) 10 9 (b) 11 9 (d) 1 9 Solutio : (a) Fom elatio 7.9, we have (b) Similaly (c) (d) MATHEMATIS

20 MODULE - I Thee is aothe elatio satisfied by Pemutatios Ad ombiatios which is also useful. We have the followig elatio: (7.10) ( 1)! ( 1)! + ( )!( 1)! ( 1)!! ( 1)!( 1)! + ()( 1)!( 1)!( 1)!( 1)! ( 1)! ( 1)!( 1)! ( 1)! ( 1)!( 1)! ( ) ( 1)! ()( 1)!( 1)! Example 7.8 Evaluate :! ( )!! (a) + (b) (c) + (d) Solutio : (a) Let us use elatio (7.10) with 7,. So, + 1 (b) Hee 9,. Theefoe, (c) Hee 6,. Theefoe, + 0 (d) Hee 11,. Theefoe, To udestad the elatio 7.10 bette, let us go back to Example 7.0 I this example let us calculate the umbe of subsets of the set, {1,,,,, 6, 7, 8, 9, 10, 11}. We ca subdivide them ito two kids, those that cotai a paticula elemet, say 1, ad those that do ot cotai 1. The umbe of subsets of the set havig elemets ad cotaiig 1 is the same as the umbe of subsets of {,,,, 6, 7, 8, 9, 10, 11} havig elemets. Thee ae 10 such subsets. 1 6 MATHEMATIS

21 Pemutatios Ad ombiatios The umbe of subsets of the set havig elemets ad ot cotaiig 1 is the same as the umbe of subsets of the set {,,,,6,7,8,9,10,11,} havig elemets. This is 10. So, the umbe of subsets of {1,,,,, 6, 7, 8, 9, 10, 11} havig fou elemets is +. But, this is as we have see i the example. So, +. The same agumet woks fo the umbe of elemet subsets of a set with elemets. This eletio was oticed by the Fech MathematiciaBlaise Pascal ad was used i the so called Pascal Tiagle, give below The fist ow cosists of sigle elemet 0 1. The secod ow cosists of 1 0 ad 1. 1 Fom the thid ow owads, the ule fo witig the eties is as follows. Add cosecutive elemets i the pevious ow ad wite the aswe betwee the two eties. Afte exhaustig all possible pais, put the umbe 1 at the begiig ad the ed of the ow. Fo example, the thid ow is got as follows. Secod ow cotais oly two elemets ad they add up to. Now, put 1 befoe ad afte. Fo the fouth ow, we have 1 +, + 1. The, we put 1 +, + 1. The we put 1 at the begiig ad the ed. Hee, we ae able to calculate, fo example, 1,., fom 0, 1, by usig the elatio The impotat thig is we ae able to do it usig additio aloe. MODULE - I The umbes occu as coefficets i the biomial expasios, ad theefoe, they ae also called biomial coefficets as we will see i lesso 8. I paticula, the popety 7.10 will be used i the poof of the biomial theoem. Example 7.9 If 10 1 fid, Solutio : Usig 10 1 o, we get MATHEMATIS 7

22 MODULE - I HEK YOUR PROGRESS 7.7 Pemutatios Ad ombiatios 1. (a) Fid the value of 1. Is 1? (b) Show that 0. Evaluate : (a) 9 (b) 1 10 (c) 1 9 (d) 1 1. Evaluate : (a) + (b) (c) + (d) +. If , fid the value of.. If 18 fid + PROBLEMS INVOLVING BOTH PERMUTATIONS AND OMBINATIONS So fa, we have studied poblems that ivolve eithe pemutatio aloe o combiatio aloe. I this sectio, we will coside some examples that eed both of these cocepts. Example 7.0 Thee ae ovels ad biogaphies. I how may ways ca ovels ad biogaphies ca be aaged o a shelf? Soluto : ovels ca be selected out of i ways. biogaphies ca be selected out of i ways. Numbe of ways of aagig ovels ad biogaphies 6 0 Afte selectig ay 6 books ( ovels ad biogaphies) i oe of the 0 ways, they ca be aaged o the shelf i 6! 70 ways. By the outig Piciple, the total umbe of aagemets Example 7.1 Fom cosoats ad vowels, how may wods ca be fomed usig cosoats ad vowels? Solutio : Fom cosoats, cosoats ca be selected i ways. Fom vowels, vowels ca be selected i ways. Now with evey selectio, umbe of ways of aagig lettes is P 8 MATHEMATIS

23 Pemutatios Ad ombiatios Total umbe of wods P MODULE - I! HEK YOUR PROGRESS Thee ae Mathematics, Physics ad hemisty books. I how may ways ca you aage Mathematics, Physics ad hemisty books. (a) if the books o the same subjects ae aaged togethe, but the ode i which the books ae aaged withi a subject does t matte? (b) if books o the same subjects ae aaged togethe ad the ode i which books ae aaged withi subject mattes?. Thee ae 9 cosoats ad vowels. How may wods of 7 lettes ca be fomed usig cosoets ad vowels?. I how may ways ca you ivite at least oe of you six fieds to a die?. I a examiatio, a examiee is equied to pass i fou diffeet subjects. I how may ways ca he fail? LET US SUM UP Fudametal piciple of coutig states. If thee ae evets ad if the fist evet ca occu i m 1 ways, the secod evet ca occu i m ways afte the fist evet has occued, the thid evet ca occu im ways afte the secod evet has occued ad so o, the all the evets ca occu i m 1 m m... m 1 m ways. The umbe of pemutatios of objects take all at a time is! P! ( )! P! The umbe of ways of selectig objects out of objects!!( )! MATHEMATIS 9

24 MODULE - I SUPPORTIVE WEB SITES Pemutatios Ad ombiatios TERMINAL EXERISE 1. Thee ae 8 tue - false questios i a examiatio. How may esposes ae possible?. The six faces of a die ae umbeed 1,,,, ad 6. Two such dice ae thow simultaeously. I how may ways ca they tu up?. A estauat has vegetables, salads ad types of bead. If a custome wats 1 vegetable, 1 salad ad 1 bead, how may choices does he have?. Suppose you wat to pape you walls. Wall papes ae available i diffet backgouds colous with 7 diffeet desigs of diffeet colous o them. I how may ways ca you select you wall pape?. I how may ways ca 7 studets be seated i a ow o 7 seats? 6. Detemie the umbe of 8 lette wods that ca be fomed fom the lettes of the wod ALTRUISM. 7. If you have widows ad 8 cutais i you house, i how may ways ca you put the cutais o the widows? 8. Detemie the maximum umbe of - lette wods that ca be fomed fom the lettes of the wod POLIY. 9. Thee ae 10 athletes paticipatig i a ace ad thee ae thee pizes, 1st, d ad d to be awaded. I how may ways ca these be awaded? 10. I how may ways ca you aage the lettes of the wod ATTAIN so that the Ts ae togethe? 11. A goup of 1 fieds meet at a paty. Each peso shake hads oce with all othes. How may had shakes will be thee.? 1. Suppose that you ow a shop which sells televisios. You ae sellig diffeet kids of televisio sets, but you show case has eough space fo display of televiso sets oly. I how may ways ca you select the televisio sets fo the display? 1. A cotacto eeds capetes. Five equally qualified capetes apply fo the job. I how may ways ca the cotacto make the selectio? 1. I how may ways ca a committe of 9 ca be selected fom a goup of 1? 1. I how may ways ca a committee of me ad wome be selected fom a goup of 1 me ad 1 wome? 60 MATHEMATIS

25 Pemutatios Ad ombiatios 16. I how ways ca 6 pesos be selected fom gade 1 ad 7 gade II offices, so as to iclude at least two offices fom each categoy? 17. Out of 6 boys ad gils, a committee of has to be fomed. I how may ways ca this be doe if we take : (a) gils. (b) at least gils. 18. The Eglish alphabet has vowels ad 1 cosoats. What is the maximum umbe of wods, that ca be fomed fom the alphabet with diffeet vowels ad diffeet cosoats? 19. Fom cosoats ad vowels, how may wods ca be fomed usig cosoats ad vowels? 0. I a school aual day fuctio a vaiety pogamme was ogaised. It was plaed that thee would be shot plays, 6 ecitals ad dace pogammes. Howeve, the chief guest ivited fo the fuctio took much loge time tha expected to fiish his speech. To fiish i time, it was decided that oly shot plays, ecitals ad dace pogammes would be pefomed, How may choices wee available to them? (a) if the pogammes ca be pefomed i ay ode? (b) if the pogammes of the same kid wee pefomed at a stetch? (c) if the pogammes of the same kid wee pefomed at a stech ad cosideig the ode of pefomace of the pogammes of the same kid? MODULE - I MATHEMATIS 61

26 MODULE - I ANSWERS Pemutatios Ad ombiatios HEK YOUR PROGRESS (a) 180 (b) 8 (c) 1 (d) 0. (a) 8 (b) 6 HEK YOUR PROGRESS (a) 1776 (b) 900. (a) 10 (b) 60. (a) (b) HEK YOUR PROGRESS (a) (i) 70 (ii) 00 (iii) 06 (iv) 1780 (v) 10 (b) (i) False (ii) False (iii) Tue (iv) False. (a) 10 (b) 00 (c) HEK YOUR PROGRESS (a) (i) 1 (ii) 10 (iii) (iv) 700 (v)! (b) (i) False (ii) Tue (iii) False (iv) False. (a) (i) 7980 (ii) 90 (b) 0 (c) 80 HEK YOUR PROGRESS (i) 00 (ii) 00 (iii) 60 HEK YOUR PROGRESS (a) (i) 86 (ii) 16 (iii) 8 (iv) 1 (b) (i) Tue (ii) False 6 MATHEMATIS

27 Pemutatios Ad ombiatios (iii) False (iv) Tue (i) 0 MODULE - I (ii) HEK YOUR PROGRESS (a), No. (a) 16 (b) 1001 (c) 71 (d). (a) 6 (b) 16 (c) 10 (d) HEK YOUR PROGRESS (a) 600 (b) MATHEMATIS 6

28 MODULE - I TERMINAL EXERISE (a) 10 (b) (a) (b) 1080 (c) 1100 Pemutatios Ad ombiatios 6 MATHEMATIS