Permutations and Combinations

Transcription

1 Pemutatios ad ombiatios Fudametal piciple of coutig Pemutatio Factoial otatio P!! ombiatio This uit facilitates you i, statig the fudametal piciple of coutig. defiig pemutatio ad combiatio. usig factoial otatio. deivig the fomula fo umbe of pemutatios ( P ).!!! deivig the fomula fo umbe of combiatios ( ). diffeetiatig betwee pemutatios ad combiatios solvig poblems usig P ad. Jacob Beoulli ( - 70 A. D.) I mathematics the at of poposig a questio must be held of highe value tha solvig it. - Geoge ato Swiss mathematicia, Jacob Beoulli has give a complete teatmet of pemutatios ad combiatios i his book As ojectadli (Posthumously published i 7 A. D)

2 UNIT- Let us begi this uit with some eal life situatios. Study them ad discuss i goups. Illustatio We see diffeet types of moto vechicles plyig o the oad. Each ad evey vehicle will have a umbe plate with it's egistatio umbe. Recall that each umbe plate will have the followig ifomatio. ) Name of the state (Eg : KA - fo Kaataka) ) Regioal taspot office code umbe. (0 - Bagaloe - Jayaaga RTO) ) Eglish alphabet/s ) Followed by a umbe (fom a sigle digit to a fou digit umbe) Do you kow how may vehicles ca be egisteed i ay oe of the seies? You kow that thee ca be 9999 vehicles!! Have you eve thought how these umbes ae geeated by the R.T.O? What mathematical piciple is behid this calculatio? Illustatio Suppose we have a suitcase with umbe lock. The umbe lock has thee wheels each labelled with te digits fom 0 to 9. The lock ca be opeed if thee specific digits ae aaged i a paticula aagemet o may be i a sequece say with o epetitios. Let us suppose that we have fogotte this sequece of digits. I ode to ope the lock, how may sequeces of thee - digits we have to check? To aswe this questio we have to check all the thee - digit umbes. But this will take lot of time. Is thee ay mathematical idea which ca help us to kow the umbe of possibilities? I this uit let us lea some basic coutig techiques by which we ae able to fid solutios fo the above situatios. As a fist step, we shall examie caefully a piciple which is most fudametal to leaig of the coutig techiques.the mai subject of this uit is coutig. Give a set of objects, the task is to aage a subset accodig to some specificatios o to select a subset as pe some specificatios. Fudametal Piciple of outig (FP) Study the followig illustatios. EV007 Example : Thee ae thee dolls as show. Two out of them have to be aaged o a shelf. I how may diffeet ways ca they be aaged? How do we cout this? Obseve the tee diagam epesetig the umbe of ways of selectig the fist doll ad the secod doll.

3 Pemutatios ad ombiatios 7 Fom this diagam, we ca coclude that the fist doll ca be selected i thee diffeet ways ad fo each of these fist selectio, the secod doll ca be selected i two diffeet ways. So thee ae totally ways. The total umbe of ways of aagig two dolls out of thee is = ways. The diffeet ways ca be epeseted as follows: Example : A boy has pats ad shits. How may diffeet pais of a pat ad a shit ca he dess up with? Thee ae two ways i which a pat ca be chose ad fo evey choice of a pat thee ae fou ways i which a shit ca be chose. Let us deote the two pats as P ad P ad the shits as S, S, S ad S. The diffeet ways of paiig a pat ad a shit ca be epeseted by a tee diagam as show below. Shits thee ae totally 8 ways. I othe wods, thee ae = 8 ways of paiig a pat ad a shit. Example : Sajay goes to a hotel to have beakfast. The meu cad idicates the followig items that ae seved: Tiffi Sweets Hot diks Idli Kesai bath offee Dosa Jamoo Tea Pooi Khaabath Pats Badam milk I how may diffeet ways ca he select oe item fom each type? Tiffi ca be chose i diffeet ways. Afte a tiffi is chose, a sweet ca be chose i diffeet ways. thee ae = 8 ways i which a tiffi ad a sweet ca be chose. P P S PS S PS S PS S PS S PS S PS S PS

4 8 UNIT- Fo each of the above 8 ways, a hot dik ca be chose i diffeet ways. thee ae 8 = ways i which Sajay ca choose the thee items. the total umbe of ways = = If we epeset tiffi as T, T, T ad T, sweets as S ad S, hot diks as H, H ad H, the possible ways ca be listed i the table as follows. T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H T S H The data fom the above thee examples is eteed i the table. Study the table ad discuss i goups to ife the piciple of coutig. Example Numbe of ways Total umbe Activity Activity Activity of ways = = 8 = We ca coclude ad state the fudametal piciple of coutig (FP) o multiplicatio piciple as follows : If oe activity ca be doe i 'm' umbe of diffeet ways ad coespodig to each of these ways of the fist activity, secod activity (idepedet of fist activity) ca be doe i '' umbe of diffeet ways the, both the activities, oe afte the othe ca be doe i (m ) umbe of ways. The above piciple ca be geealised fo ay fiite umbe of evets. Fo thee activities, the FP ca be stated as: If oe activity, ca be doe i 'm' umbe of diffeet ways, fo each of these 'm' diffeet ways, a secod activity ca be doe i '' umbe of diffeet ways ad fo each of these activities, a thid activity ca be doe i 'p' ways, the all the thee activities oe afte the othe ca be doe i (m p) umbe of ways. We ca ow easily fid the total umbe of ways of doig moe tha two activities without actually listig, coutig o dawig tee diagam. We shall illustate this FP with a few moe examples. ILLUSTRATIVE EXAMPLES Example : How may - digit umbes ca be fomed fom the digits {,,,, } without epetitio ad with epetitio? Sol: The digits i the selectio set ae {,,,, }. Sice we have to fom - digit umbe, let us daw boxes, oe fo uits place ad aothe fo tes place. T U

5 Pemutatios ad ombiatios 9 We fill the uits place fist ad the tes place. The box fo uits place ca be filled i diffeet ways with the digits,,, ad. Afte fillig the uits place we ae left with digits, as epetitio of digits ae ot allowed, the box fo tes place ca be filled i diffeet ways. By applyig fudametal piciple of coutig we get = 0 ways. thee ae 0 ways of fomig - digit umbes by usig,,,, without epetitio. If the epetitio of digits ae allowed the the tes place ca also be filled by diffeet ways, usig,,, ad. The total umbe ways a - digit umbe ca be fomed by usig,,, ad with epetitio is = ways. Note: Discuss i goups ad veify the aswe by actually listig the - digit umbes. Example. How may lette code ca be fomed by usig the five vowels without epetitios? Sol. The vowels ae a, e, i, o ad u. We have to fom a thee lette code. The fist lette ca be chose i ways. The secod lette ca be chose i ways. The thid lette ca be chose i ways. ( o epetitio is allowed) ( o epetitio is allowed) The total umbe of ways i which the lette code ca be fomed without epetitio = = 0 ways Example. How may - digit umbes ca be fomed fom the digits 0,,, ad with epetitios? Sol. A Thee - digit umbe will have thee places - Hudeds, tes, uits. The digits i the selectio set ae {0,,,, } Hudeds place - Zeo caot occupy hudeds place, because it becomes a - digit umbe. Hece, the hudeds place ca be filled i by ways, i.e.,,,. Tes place - We have five ways of fillig tes place, as epetitios ae allowed ad zeo ca also be used. Uits place - This ca also be filled i diffeet ways. the total umbe of - digit umbes that ca be fomed usig 0,,,, ae = 00 umbes. Example. Now let us ty to solve the suitcase lock poblem (illustatio i page ), by applyig the fudametal piciple of coutig. Sol. The umbe lock of the suitcase has thee digits. Each umbe ca be fomed i 0 diffeet ways usig the digits 0,,,,,,, 7, 8,

6 70 UNIT- It is also evidet that the digits ca be epeated. the total umbe of ways the umbe lock code ca be fomed is = 000 ways So, to ope the suitcase 000 diffeet ways have to be checked. EXERISE.. How may - digit umbes ca be fomed usig the digits,,,,, without epeatig ay digit.. How may digit eve umbes ca be fomed usig the digits,, 7, 8, 9, if the digits ae ot epeated?. How may lette code ca be fomed usig the fist 0 lettes of Eglish alphabet, if o lette ca be epeated?. How may digit telephoe umbes ca be fomed usig the digits 0 to 9, if each umbe stats with ad o digit appea moe tha oce?. If a coi is tossed times, fid the umbe of outcomes.. Give flags of diffeet colous, how may diffeet sigals ca be geeated if each sigal equies the use of flags oe below the othe? Pemutatios ad ombiatios While leaig the fudametal piciple of coutig, we have obseved that i all the examples some umbe of objects ae give ad out of these a few objects ae to be chose. Now, let us study moe about the selectio ad aagemet of thigs. Study the examples give i the table. Read the fist poblem i colum 'P' ad the the fist poblem i colum '' Now compae them. Repeat the same pocedue fo the emaiig set of poblems. Sl No.... olum 'P' pesos cotest fo the post of a pesidet ad a secetay. I how may ways they ca be elected? A child has 8 focks ad pais of shoes. I how may ways the child ca dess heself? I how may ways ca all the lettes of the wod TEAH be aaged? olum '' Thee ae codidates cotestig fo posts. I how may ways ca the posts be filled. Thee ae 8 white ad ed oses i a gade. I how may ways flowes of which ae ed be plucked? I how may ways all the lettes of the wod MEANS ca be selected? Do you otice ay similaties i the poblems of each colum? Fo the poblems i the colum 'P', we have to fid out the umbe of ways of aagemets. Fo the poblems i the colum '' we have to fid the umbe of ways of selectios.

7 Pemutatios ad ombiatios 7 I solvig polems of the above types the fist step is we must caefully idetify, whethe it is the aagemets o selectios. As a example let us udestad the fist poblem i the above table. Let the pesos be A, B ad. Thee ae two posts, pesidet ad a secetay. Let us have the box otatio ad wite all the possible ways. Pesidet A A B B Secetay ase. If 'A' is the pesidet the 'B' ca be secetay o '' ca be secetay ase : If 'B' is the pesidet the 'A' ca be secetay o '' ca be secetay ase : If '' is the pesidet the 'A' ca be secetay o B ca be the secetay. So all togethe thee possible ways of electig. B A A B Let the pesos be A, B ad. Thee ae two posts oly. Let us have the box otatio ad wite all the possibilities. hoices fo posts SL No. Post - Post - A A B ase. If 'A' is selected fo oe post, aothe post ca be give to B o. ase : If 'B' is selected fo oe post the the othe post ca be give to but ot to A because BA ad AB ae the same selectios. Hece thee ca be ways by which the posts ca be filled up B It is also evidet fom the above discussio that, i examples of colum P the aagemet of thigs o objects o pesos is doe with egad to ode. I examples of colum, the selectio of thigs o objects o pesos is doe without egad to ode. Such aagemets with egad to ode ae called pemutatios ad selectios without egad to ode ae called combiatios. Pemutatio : A pemutatio is a odeed aagemet of a set of objects. It is a act of aagemets of objects i a odely way. A pemutatio of elemets take at a time is ay odeed subset of elemets take fom the set of elemets. The umbe of pemutatios of '' diffeet objects take '' at a time is deoted as P whee. ombiatio : A combiatio is a selectio of a set of objects without ay ode. It is a act of selectio of objects ot ivolvig ay odely way. A combiatio of elemets take at a time is ay subset of elemets take fom the set of elemets (without egad to ode). Note that i a set thee is o ode fo listig the elemets. e.g. {,, } = {,, } The umbe of combiatios of '' diffeet objects take '' at a time is deoted as, whee.

8 7 UNIT- EXERISE. I. Below ae give situatios fo aagemets ad selectios. lassify them as examples of pemutatios ad combiatios.. A committee of membes to be chose fom a goup of people.. Five diffeet subject books to be aaged o a shelf.. Thee ae 8 chais ad 8 people to occupy them.. I a committee of 7 pesos, a chaipeso, a secetay ad a teasue ae to be chose.. The owe of childe's clothig shop has 0 desigs of focks ad of them have to be displayed i the fot widow.. Thee-lette wods to be fomed fom the lettes i the wod 'ARITHMETI'. 7. I a questio pape havig questios, studets must aswe the fist questios but may select ay 8 of the emaiig oes. 8. A ja cotais black ad 7 white balls. balls to be daw i such a way that ae black ad is white. 9. Thee-digit umbes ae to be fomed fom the digits,,, 7, 9 whee epetitios ae ot allowed. 0. Five keys ae to be aaged i a cicula key ig.. Thee ae 7 poits i a plae ad o of the poits ae colliea. Tiagles ae to be daw by joiig thee o-colliea poits.. A collectio of 0 toys ae to be divided equally betwee two childe. I each of the above examples give easo to explai why it is pemutatio o combiatio. Ty: List atleast examples each fo pemutatio ad combiatio. Discuss i the class. To fid the geeal fomula fo the umbe of ways by which th box ca be filled up: By usig FP we ca wite the expasio of each pemutatio as follows :- P P P P P I each pemutatio, obseve caefully the umbe of ways i which the last box is filled up. Thee is some commo patte i all the pemutatios. Thee is a elatioship betwee values of, ad the umbe of ways i which the last box ca be filled up. The elatio is as follows :

9 Pemutatios ad ombiatios 7 'Subact '' fom '' ad add oe to it' to get the umbe of ways by which th box ca be filled up i.e., ( + ) Note that whe use say th box, is atleast. So > 0 Study the followig examples: ) oside P - we kow that the last box i.e., th box ca be filled up i just way. We ca get this by subtactig ( fom ) ad add to it. i.e., ( + ) = 0 + = way. ) oside P - Hee also we kow that the last box i.e., th box ca be filled up i ways. We ca get this by subtactig ( fom ) ad add to it. i.e., ( + ) = + = ways. ) oside P - Hee also we see fom the above table last box ca be filled up i ways. This we ca get by subtactig ( fom ) ad the add to it. i.e., ( + ) = + = ways Hece, we ca coclude that i P the th box ca be filled i ( + ) ways. To fid the umbe of pemutatios of '' distict objects take '' at a time whee 0 ad the objects do ot epeat. The umbe of pemutatios of '' objects take '' at a time is same as the o of ways of fillig '' blak boxes with '' give objects. oside '' diffeet objects ad '' blak boxes as show below. ( ) ( ) [ )] [ )] ways ways ways ways ways The fist box ca be filled by '' umbe of ways by puttig '' diffeet give objects. Thus, thee ae '' diffeet ways of fillig up the fist box. Afte fillig up the fist box by ay oe of the '' objects, we ae left with ( ) objects as epetitios ae ot allowed. So, the secod box ca be filled up i ( ) umbe of ways. Thus, the fist boxes ca be filled up i ( ) umbe of ways (by FP). Now afte fillig up the fist boxes, we ae left with ( ) objects. Thus, the d box ca be filled up i ( ) umbe of ways. Agai by F.P. the fist thee boxes ca be filled up i ( ) ( ) ways. Obseve that a ew facto is itoduced with each ew box filled up, ad that at ay stage the umbe of factos is the same as the umbe of boxes filled up. Hece, by FP the umbe of ways of fillig '' boxes i successio is give as follows. ( ) ( )... [ ( )]

10 7 UNIT- Thus the umbe of pemutatios of '' objects take '' at a time is give by P P = ( ) ( )... [ ( )] oollay : The umbe of pemutatios of '' objects take all at a time is P = ( ) ( )... [ ( )] P = ( ) ( )... o P = ( ) ( )... Factoial otatio : ('' factos) Obseve that the RHS of the above expesssio is the poduct of fist atual umbes. It is deoted by a otatio!.! is ead as factoial. ( ) ( )... =!! deotes the poduct of fist atual umbes. Thus! = o Remembe :! = o! = o! =... o ( ) ( )... 0! by defiitio is take as. [Note : If is a egative umbe o a decimal,! is ot defied.]! is the poduct of fist '' atual umbes.! = ( ) ( )... S is the sum of fist '' atual umbes. S = Study the expasios of factoial otatios give below.! = ( ) ( )... ( )! = ( ) ( ) ( )... ( + )! = ( + ) ( ) ( )... ( )! = ( ) ( ) ( )... ( )! = ( ) ( ) ( )... I geeal ;! = ( ) ( )...! = [( ) ( )... ]! = [( )!] o! = ( ) ( )! o! = ( ) ( ) ( )! ad so o. Fo example, oside!! =! = (!)! = (!)! = (!)

11 Pemutatios ad ombiatios 7 EXERISE.. ovet the followig poducts ito factoials. (i) 7 (ii) (iii) (iv) 8. Evaluate : (i)! (ii) 9! (iii) 8!! (iv) 7!! (v)! 9!! (vi) 0! 8!. Evaluate : (i)!! (ii)!!! whe = ad =. Fid the LM of!,!,!. (i) If ( +)! = ( )! fid the value of (ii) If 9! 0!! fid the value of. Simplify : ( )! ( )! ( ) To deive the fomula fo P i factoial otatio By Fudametal Piciple of coutig, P = ( ) ( )... ( + ) oside the RHS of the equatio, ( ) ( )... ( + ) The fist facto of the RHS is. The last facto of the RHS is ( + ). RHS is the poduct of atual umbes statig fom '' i descedig ode upto ( + ). Had the poduct cotiued upto what we would have got? We would have got! What exta facto we have to wite to get!? [( ) ( )... ] which is ( )! P = ( ) ( )... ( + ) P = ( ) ( )... ( ) ( ) ( )... ( ) ( )... P =!! P =!! [If > 0] Numbe of pemutatios of '' thigs take '' at a time is The above fomula fo P holds good oly whe epetitios ae ot allowed. P!!, whee 0.

12 7 UNIT- The umbe of pemutatios of '' diffeet objects take '' at a time, whee epetitio is allowed is. Now let us study some special cases of P ase : What happes whe = 0? We shall see it ow. If = 0 i how may ways ca we aage 0 objects fom objects? Aagig o object at all is the same as leavig behid all the objects as they ae ad we kow that thee is oly oe way of doig it. Thus P0 P = o! 0! =!! = Example : 00 P, 0 P, 98 P ase : Let = P = ( ) ( )... ( + ) P = ( ) ( )... ( + ) P = ( ) ( )... P = ( ) ( )... P =! Also, P!!!!! 0! (ecall the defiitio 0! = ) Example : 009 P 009 = 009!, 0 P 0 = 0!, 00 P 00 = 00! ase : Let = P!! P!! P!! P = Example : 00 P = 00, 7 P = 7 Discuss: P is meaigless. Why? Remembe: P!! If = 0 the P 0 = If = the P =! If = the P =

13 Pemutatios ad ombiatios 77 ILLUSTRATIVE EXAMPLES Example. Evaluate : (i) 7 P (ii) 8 P Sol. (i) 7 P 7! 7! 7! 7!!! 7 0 (ii) 8 P 8! 8! 8 7! 8!!! Example. Fid '', if P = P Sol.!!!!!!!!!! ( ) ( ) = + = 0 ( 8) ( ) = 0 = Example. Pove that! + ( + )! =! ( + ) [ = 8 has o meaig] Sol. LHS:! + ( + )! =! + ( + )! =! ( + + ) =! ( + ) = RHS Example. Fid '' if P P Sol. ( ) ( ), ( ) = 0 = = 0 = 0 Example. If + P : P = : fid. Sol. P P i.e., + P = P.!.!!!!!!! 0!!

14 78 UNIT- 0 ( + ) = ( + ) ( + ) = 0 ( + ) ( ) = 0 o =. Sice is a positive itege, =. Example. If P = 00, fid '' Sol. P = 7 = 00! = 7! = 7 (ii) If P = 90 fid '' Sol. P = 90 ( ) = 0 9 = 0 (iii) If P = 990 fid '' Sol. P = 990 P = 0 9 = EXERISE.. Evaluate: i) P ii) 7 P iii) 8 P 8 iv) P v) 8 P 0.. If P = 0 P fid ''. If P = + P fid ''. If P =. P fid ''.. If P : P = : fid ''. If P : + P = : fid ''.. If 9 P + 9 P = 0 P, fid ''. If 0 P + : P = 0 :, fid ''.. Pove that + P + = ( + ) P. Show that 0 P = 9 P + 9 P

15 Pemutatios ad ombiatios 79 Pactical Poblems o Pemutatio Example. sogs ae to be edeed i a pogamme. I how may diffeet odes could they be pefomed? Sol. The diffeet odes i which sogs that ca be pefomed is P =!! = [ FP] o = 70 diffeet odes P!!! P! = = 70 sogs ca be edeed i a pogamme i 70 diffeet odes. Example. How may wods (with o without dictioay meaig) ca be made fom the lettes i the wod LASER assumig that o lette is epeated it, such that (i) All lettes ae used at a time (ii) lettes ae used at a time (iii) All lettes ae used such that it should begi with lette A ad ed with lette R Sol. Numbe of lettes i the wod LASER is. (i) If all lettes ae used at a time, the =, = The umbe of wods that ca be fomed P =!!!! 0!! 0 0 wods ca be fomed with o without meaig. (ii) If lettes ae used at a time, the =, = The umbe of wods fomed with o without meaig P =!! 0 (iii) So thee will be 0 wods with o without meaig. Box Box Box Box Box lette A is fixed lette R is fixed Thee ae lettes i the give wod LASER of which two lettes A ad R ae fixed i the fist & last place espectively. So oly thee boxes have to be filled amely d, d ad th. This ca be doe i P!! 0! ways Thee will be oly wods with o without meaig which begis with lette 'A' ad ed with the lette 'R'.

16 80 UNIT- Example. I how may ways ca 7 diffeet books be aaged o a shelf? I how may ways thee paticula books ae always togethe? Sol. The 7 books ca be aaged i 7 P 7 = 7! = 7 = 00 ways. Sice thee paticula books ae always togethe, let us tie thee books togethe ad the coside them as oe book (o oe uit). Remaiig fou books have to be cosideed sepaately. So i all we ca coside 7 books as [ books + book ] = books ( uit of books) These books ca be aaged i P =! = = 0 ways. I each of these 0 ways, the thee paticula books ca be aaged i! = = ways { (B B B )(B B B ), (B B B ), (B B B ), (B B B )(B B B ) By FP, the total umbe of aagemets P! = 0 = 70 ways. So, the umbe of ways of aagig 7 books, so that thee paticula books is eve togethe is = 0 Example. How may five digit umbes ca be fomed with the digits {,,, 7, 9} which lie betwee 0,000 ad 90,000 usig each digit oly oce. Sol. Te thousad Thousad Huded Teth uit 's place place place place place ways {, o 7} ways ways ways way Sice we equie umbes which ae geate tha 0,000 ad less tha 90,000, the te thousadth place ca be filled by {, o 7} oly. Theefoe te thousadth place ca be filled i by ways oly. The emaiig fou places ca be filled with the emaiig fou digits. This ca be doe is! ways. Theefoe the total umbe of umbes lyig betwee 0,000 ad 90,000 is! { FP} = = 7 Example. A DNA molecule will have a itoge base which cosists of diffeet bases A, G, T o all attached to it? I how may ways the thee bases ca be aaged without epetitio. Sol. Hee, = ad =. Hece the umbe of ways A = Adeie = ytosie G = Geaie T = Thymie P! ways!

17 Pemutatios ad ombiatios 8 EXERISE.. How may wods with o without dictioay meaig ca be fomed usig all the lettes of the wod 'JOULE' usig each lette exactly oce?. It is equied to seat me ad wome i a ow so that the wome occupy the eve places. How may such aagemets ae possible?. I how may ways ca wome daw wate fom wells, if o well emais uused?. Seve studets ae cotestig the electio fo the Pesidetship of the studet's uio. I how may ways ca thei ames be listed o the ballot papes?. 8 studets ae paticipatig i a competitio. I how may ways ca the fist thee pizes be wo?. Fid the total umbe of - digit umbes. 7. Thee ae stickes of diffeet sizes. It is desied to make a desig by aagig them i a ow. How may such desigs ae possible? 8. How may - digit umbes ca be fomed usig the digits,,, 7, 8, 9 (epetitios ot allowed)? (a) How may of these ae less tha 000? (b) How may of these ae eve? (c) How may of these ed with 7? 9. Thee ae buses uig betwee two tows. I how may ways ca a ma go to oe tow ad etu by a diffeet bus? To fid out the umbe of combiatios of '' distict objects take '' at a time whee 0 oside the followig examples of selectig ad aagig lettes fom a, b, c ad d. Hee = : ombiatios Pemutatios =, a, b, c, d a, b, c, d = P = Each combiatio of give ise to! Pemutatio. = ab,ac,ad,ba,bc,bd ab,ac,ad,bc,bd,cd ca,cb,cd,da,db,dc = P = Each combiatio of give ise to! Pemutatios. = abc,abd,acd,bcd abc abd bcd acd = acb adb bdc adc bac bad cbd cad bca bda cdb cda cab dab dbc dac cba dba dcb dca P = ways Each combiatio of give ise to! Pemutatios.

18 8 UNIT- = abcd abcd bacd cabd dabc = abdc badc cadb dacb acbd bcad cdba dbac acdb bcda cdab dbca adcb bdac cdab dcab abdc bdca cbda dcba P : I Geeal: Each combiatio i gives ise to! Pemutatios. Each combiatio i give ise to! pemutatios. How may pemutatios ca thee be altogethe? Total o of Pemutatios =! P =! =! i.e. =!!! Theefoe, the umbe of combiatios of '' distict objects, take '' at a time is give by P!! -! Note: The elatioship betwee P ad is give by = P! Study the vaious expasios of.. P!!!!....!!....! Now let us coside some special cases of ase (i): Let = 0!! If = 0 the, 0 0! 0! 00 0 =, 00 0 =, = 0 =

19 Pemutatios ad ombiatios 8 ase (ii): Let = If = the,!!! =!! = Example: 00 = 00, 00 = 00 7 = 7 ase (iii): Let = If = the,!!!!!0!!! = Example: =, =, = Remembe :!!! If = 0, 0 =, If =, =, If =, = ILLUSTRATIVE EXAMPLES Example. Evaluate : (i) (ii) 8 Sol.!!! (i)!!!!!!!! (ii) 8 8! 8! 8!!!! 8 7!! Example.If = 0, fid Sol. = 0!!!!!!!!! 0 ( ) = 0 ( ) = 0 = 0 =

20 8 UNIT- Example. If P = 0 ad =, fid ''. Sol. P =! 0 =! 0! = =! = Example. Pove that = Sol. We kow that!!! Replacig by ( ) i the equatio (i) we get!!!...(i)!!!!!!...(ii) By compaig (i) ad (ii) we ca say that = Example. If 8 =, fid. Sol. 8 = 8 = 8 = = + 8 = 0. EXERISE.. Evaluate (i) 0 (ii) 0 0 (iii) (i) If = 7 fid. (ii) If P = 80, =, fid.. If : = :, fid.. Veify that = 9. Pove that whe.. If - : : + = : :, fid.

21 Pemutatios ad ombiatios 8 Example. A ma has fieds. I how may ways ca he ivite oe o moe of them to a paty? Sol. The diffeet ways of combiatios of ivitig fieds ae ivitig exactly oe,, exactly,... exactly. These ca be doe i,,,,, ways. The total umbe of ways i which oe o moe ca be ivited = = 0 = = ways. Example. Fo a set of tue o false questios o studet has witte all coect aswes ad o two studets have give the same sequece of aswes. What is the maximum umbe of studets i the class fo this to be possible? Sol. Each questio ca be asweed i exactly ways eithe T o F. Total umbe of ways i which the questios ca be asweed = = Out of these ways exactly oe will be all coect aswe. Sice o studet has witte all coect aswes The maximum possible umbe of studets = =. Example. fieds shake hads mutually. Fid the umbe of hadshakes. Sol. Let the fieds be A, B, ad D. Fo oe had shake, thee will be pesos ivolved. To stat with let peso 'A' shakes had with the peso 'B'. It is the same as peso 'B' shakig hads with peso 'A'. AB ad BA ae the same. So i fidig the umbe of hadshakes, we obseve that ode is ot impotat. Hece the umbe of hadshakes will be always A {AB, A, AD} B D B, ad =. D I this poblem =. Numbe of hadshakes will be. {B, BD}!!! Example. Eveybody i a fuctio shakes had with eveybody else. The total umbe of hadshakes is. Fid the umbe of pesos i the fuctio. Sol. Let the umbe of pesos i the fuctio be. The = {D} D!!!

22 8 UNIT- ( ) = 90 ( ) = 0 9 ( ) = 0(0 ) = 0 Hece the umbe of pesos i the fuctio is 0. Example. How may committees of five with a give chaipeso ca be selected fom pesos? Sol. The chaipeso ca be choose i ways ad followig this the othe fou o the committee ca be choose i ways. The possible umbe of such committees = = 0 = 90. Example. Thee ae 8 poits such that ay of of them ae o colliea. How may staight lies ca be daw by joiig these poits? Sol. A staight lie is obtaied by joiig poits. Let the poits be A ad B. The staight lie is obtaied by joiig eithe A with B o B with A. AB ad BA ae same. This is a poblem o combiatio. Total umbe of lies that ca be daw out of '' o-colliea poits = Veificatio : We kow,!!! Hee, = 8, = 8 8! 8 = 8!! 8 8 = 8 8 7! 8 staight lies ca be daw by joiig 8 poits.! Example 7. Thee ae 0 poits such that ay of them ae ocolliea. How may tiagles ca be fomed by joiig these poits? Sol. A tiagle is fomed by joiig o-colliea poits. Hee the ode i which the poits ae joied is ot at all impotat. A Total umbe of tiagles that ca be daw out of '' o-colliea poits = B 8 8 Hee, = 0, =,!!! 0 0! 0! 0!! 7!! ! 7! = 0 0 tiagles ca be fomed by joiig poits out of 0 o-colliea poits.

23 Pemutatios ad ombiatios 87 Alteate method:!!!!!!!! If = 0 A = 0 B Example 8. How may diagoals ca be daw i a hexago? Sol. A hexago has vetices = A diagoal is obtaied by joiig the opposite vetices i pais. Total umbe of sides ad diagoals = = lies icludes sides. Numbe of diagoals = = 9 Alteate method: Numbe of diagoals = Numbe of staight lies fomed Numbe of sides of polygo = = = Numbe of diagoals i a polygo of sides = I a hexago, = Numbe of diagoals Example 9. The maximum umbe of diagoals i a polygo is. Fid the umbe of sides. Sol. Numbe of diagoals i a polygo = 9 ( ) = ( ) = 7 ( ) = 7(7 ) = 7

24 88 UNIT- Alteate : ( ) = 8 = = 0 ( 7) + ( 7) = 0 ( 7) ( + ) = 0 7 = 0 OR + = 0 = 7 = '' caot be a egative umbe Hece = 7 (Heptago) Remembe Whe '' umbe of o-colliea poits ae give i a plae, * umbe of staight lies = * umbe of tiagles = * umbe of diagoals i a polygo = Pascal Tiagle Pascal, Blaise ( - ) geat Fech Mathematicia, pobabilist, combiatioist, physicist ad philosophe. Obseve the tiagle patte discoveed by Pascal. ompae the coespodig elemets i the two tiagles. The tiagle o the ight side is called Pascal tiagle

25 Pemutatios ad ombiatios 89 EXERISE.7. Out of 7 osoats ad vowels, how may wods of cosoats ad vowels ca be fomed?. I how may ways ca spotsme be selected fom a goup of 0?. I how may ways a cicket team of eleve be selected fom 7 playes i which playes ae bowles ad the cicket team must iclude bowles?. How may (i) lies (ii) tiagles ca be daw though 8 poits o a cicle?. How may diagoals ca be daw i a (i) decago (ii) icosago. A Polygo has diagoals. Fid the umbe of sides. 7. Thee ae white ad ed oses i a gade. I how may ways ca flowes of which ed be plucked? 8. I how may ways ca a studet choose couses out of 9 couses, if couses ae compulsoy fo evey studet? 9. Thee ae questios i a questio pape. I how may ways ca a boy solve oe o moe questios? 0. I how may ways cads fom a pack of playig cads ca be chose?. I a electio thee ae 7 cadidates ad thee ae to be elected. A vote is allowed to vote fo ay umbe of cadidates ot geate tha the umbe to be elected. I how may ways ca we vote? EXERISE.8. If thee ae peiods i each wokig day of a school, i how may ways ca oe aage subjects such that each subject is allowed at least oe peiod?. A committee of is to be fomed out of me ad ladies. I how may ways ca this be doe whe (i) at least ladies ae icluded. (ii) at most ladies ae icluded.. A spots team of studets is to be costituted choosig at least fom class IX ad at least fom class X. If thee ae 8 studets i each of these classes, i how may ways ca the team be costituted?. I a vegetable mela o fai, a atist wats to make mascot (a toy that epesets a ogaizatio) with Beas, aot, Peas ad Tomato i a lie. I how may ways ca the atist make the mascot?. Fom a goup of studets, 8 ae to be chose fo a excusio. Thee ae studets who decide that eithe of them will joi o oe of them will joi. I how may ways ca the 8 be chose?. How may chods ca be daw though 0 poits o a cicle? 7. The Eglish alphabet has vowels ad cosoats. How may wods with diffeet vowels ad diffeet cosoats ca be fomed fom the alphabet? 8. I how may ways lettes be posted i post boxes, if ay umbe of lettes ca be posted i all of the thee post boxes? 9. I a goup of studets thee ae scouts. I how may ways ca studets be selected, so as to iclude at least scouts?

26 90 UNIT- Pemutatio ad ombiatio Fudametal Piciple of outig (FP) Pemutatio P ombiatio Factoial otatio!! P = ( )! whee 0 =!!( )! P =! If = 0, P 0 = If =, P =! If =, P = If = 0, 0 = If =, = If =, = ANSWERS EXERISE. ] 0 ] ] 70 ] ] 8 ] 0 EXERISE. ] (i) 7! (ii) 8! (iii) 9! (iv)! ] (i) 70 (ii),,880 (iii) 0,00! (iv) (v) 0 (vi) 870 ] (i) 0 (ii) 0 ] 70 ] (i) (ii) ] ( + ) EXERISE. ] (i),880 (ii),0 (iii) 0,0 (iv) (v) ] (i) 7 (ii) (iii) ] (i) (ii) 8 ] (i) (ii) EXERISE. ] 0 ],880 ] 70 ] ] ] 90 7] 0 8] (a) 80 (b) 0 (c) 0 9] 0 EXERISE. ] (i) 0 (ii) (iii),,700 ] (i) (ii) 7 ] ] 7 EXERISE.7 ],00 ] ],00 ] (i) 8 (ii) ] (i) (ii) 70 ] 7] 8 8] 9] 0],70,7 ] EXERISE.8 ],00 ] (i) 8 (ii) 8 ], ] ] 7 ] 90 7] 0,00 8] 9]