CISC 203: Discrete Mathematics for Computing II Lecture 2, Winter 2019 Page 9

Transcription

1 Lectue, Wte 9 Page 9 Combatos I ou dscusso o pemutatos wth dstgushable elemets, we aved at a geeal fomula by dvdg the total umbe of pemutatos by the umbe of ways we could pemute oly the dstgushable elemets. We dd so ode to avod ovecoutg detcal pemutatos. If we exted the dea of dstgushablty to mea dstgushable up to odeg, the we obta a ew coutg techque whee we cae oly about the umbe of elemets we tae ad ot the ode whch those elemets ae aaged. Istead of pemutatos of elemets, we ae tag combatos of elemets.. Defto We ca defe the oto of a combato fomally, just le we defed pemutatos fomally the pevous secto. Both pemutatos ad combatos ely o sets, so we eque a lttle set theoy owledge. Howeve, you ve lely ow what combatos wee eve sce you fst leaed about sets: combatos ae just subsets dsguse. Defto (-combato). Gve a set A, a -combato of A s a sze- subset of elemets fom A. Sce combatos ae subsets, ad sce the aagemet of elemets a set does t matte, the aagemet of elemets combatos does t matte. Ths s the ey dstcto betwee pemutatos ad combatos that we alluded to the pevous secto: odeg mattes fo pemutatos, but ot fo combatos. Futhe ote that we always use the tem -combato whe efeg to a specfc value o calculato. Thee s o such thg as a combato of a set, sce f we followed the same dstcto betwee pemutato ad -pemutato, a combato of a set would just be the set tself. Thus, whe we use the wod combato o ts ow, we mea t the boad, o-fomal sese of selectg elemets fom a set wthout odeg. Example. Suppose we have a set A {,,, }. All of the possble -combatos of A ae {, }, {, }, {, }, {, }, {, }, ad {, }; exactly the same as all of the possble sze- subsets of A. It s coect fo us to cosde both {, } ad {, } to be -combatos of A, sce they both cota the same subset of elemets fom A. Theefoe, we oly cout that patcula -combato oce. Now that we ow what combatos ae, how ca we cout all possble -combatos of a -elemet set? Ths poblem souds vey smla to ou pevous poblem of coutg all -pemutatos of a -elemet set; deed, we ca tae almost the exact same appoach we too whe coutg -pemutatos. The oly d eece s that, sce odeg does t matte wth combatos, we eed to clude oe addtoal tem to guad agast ovecoutg. Theoem. The umbe of -combatos of a set wth elemets, whee apple apple, s C(, )!( )!. Poof. We ca cout the umbe of -combatos of a set wth elemets by fst calculatg the umbe of -pemutatos of the same set, ad the dvdg by the umbe of pemutatos of a -elemet set. The dvso s ecessay because the odeg of the elemets does ot matte. Thus, we have P (, ) C(, ) P (, ) /( )!!/( )!!( )!.

2 Lectue, Wte 9 Page Note that, as a cosequece of Theoem, we have C(, ), C(, ), ad C(, ) fo all. The values C(, ) ad C(, ) should mae sese fom a set-theoetc stadpot, sce fo ay -elemet set A, thee s oly oe zeo-elemet subset (;) ad oly oe -elemet subset (the set A tself). Rema. You mght have otced that pat of the pevous poof looed smla to the poof of Theoem. Ths was ot by cocdece; t s possble to pove Theoem usg C(, ) stead of P (, ). Ty t! Befoe we cotue, t s wothwhle to pot out a teestg symmety popety of combatos that may help us to solve some poblems. I a -combato, we tae elemets fom a -elemet set. Howeve, ths s o d eet fom us tag ( ) elemets ad leavg them out of ou fal choce. Fom ths obsevato, we get the afoemetoed popety. Theoem 7. Fo all atual umbes ad, whee apple apple, C(, ) C(, ). Poof. Recall that C(, )!( )!.Substtutg( ) fo, we get C(, ) ( )!( ( ))! ( )!!, ad hece C(, ) C(, ). A commo questo to hea fom studets by ths pot s how ca we tell whethe we eed to calculate pemutatos o combatos a poblem? It s cetaly a easoable questo to as, ad you wo t be faulted fo wodeg ths youself. Ufotuately, thee s o suefe tc fo detemg whch coutg techque to use, apat fom detemg whethe the poblem statemet emphaszes odeg of elemets. Thus, f a questo ased I how may ways ca we choose faculty membes fom a depatmet of faculty membes? the we would use combatos, sce we cae about oly the umbe of faculty membes. If, o the othe had, a questo ased I how may ways ca we le up faculty membes fo a depatmet photo? the we would use pemutatos, sce we cae about both the umbe ad the odeg of faculty membes. Whe doubt, emembe ths memoc: combatos ae fo choosg some umbe of elemets, ad pemutatos ae fo placg those elemets a specfc ode. Example 8. A goup of fve studets ae beggg the dscete mathematcs stucto to come up wth examples that do t volve wtg o schedulg exams. If the goup chooses thee epesetatves to tal wth the stucto dug o ce hous, how may possble combatos of epesetatves ae thee? Suppose we call the studets Alce, Bob, Caol, Davd, ad Eve. If, fo stace, Alce, Bob, ad Caol atted the o ce hous, the that combato s o d eet tha f Caol, Bob, ad Alce atted; the epesetatves ae the same. Altogethe, we have the followg subsets of thee epesetatves each: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. I othe tems, we have that C(, )!!( )!!!!. Example 9. Recall that a byte cossts of eght bay dgts. Call a byte balaced f t cotas a equal umbe of bts ad bts. How may balaced bytes exst? We ca fame ths poblem the followg way. Sce a balaced byte cotas a equal umbe of s ad s, we must have fou occueces of each bt wth a balaced byte.

3 Lectue, Wte 9 Page Thus, f we stat wth a bla byte wth eght spaces, ad we fll fou of those spaces wth s, the we ae foced to fll the emag fou spaces wth s. I how may ways ca we fll fou spaces wth s? Ths questo s equvalet to asg how may ways we ca choose fou spaces out of eght. Ths gves C(8, ) 8!!(8 )! 8!!! 7, so thee exst a total of 7 balaced bytes.. Combatos wth Repetto Just le wth pemutatos, t s possble fo us to calculate the umbe of -combatos of a set whe we ae able to select elemets fom the set moe tha oce. Ule -combatos wthout epetto, hee we ae able to select moe copes of elemets ou -combato tha those that ae the ogal set. Thus, a -combato wth epetto s ot ecessaly a subset of the ogal set. How do we llustate the pocess of tag a -combato wth epetto? Istead of us tag ad eplacg elemets of the set to fom the -combato, we wll wo evese by wtg out how may copes of each elemet ou -combato wll cota. We epeset ths sceao a uquely Ameca style: usg the so-called stas ad bas method. (Iteestgly, the mathematca who populazed ths method Wllam Felle was bo Coata, ot Ameca.) Wth the stas ad bas method, we patto ou set to classes usg bas (deoted ). Each class coespods to a dstct elemet the set. We the epeset the umbe of copes of elemets that class to be cluded ou -combato usg the appopate umbe of stas (deoted F). Example. A studet s eollg couses fo the upcomg academc yea. They pla to eol couses. If the couses ae to be selected fom the set {CISC, MATH, PHYS, BIOL, CHEM}, the we ca patto these fve classes of couse codes usg fou bas: {z} CISC MATH PHYS BIOL CHEM The studet wats to tae fve CISC couses, thee MATH couses, oe PHYS couse, ad oe CHEM couse. If we deote oe couse by oe sta, ou stas ad bas dagam wll loo le the followg: FFFFF FFF F F As the pevous example llustates, the umbe of elemets the set ad the umbe of selectos we mae dctate the umbe of stas ad bas that ae avalable to us. If ou set cotas elemets ad we wsh to mae j selectos, the we wll have j stas ad ( ) bas. As we also saw the pevous example, t s pefectly fe to have zeo stas a gve patto; ths just meas we made o selectos of elemets fom the coespodg class. Usg stas ad bas ths way, t becomes evdet that the umbe of ways to fom a -combato wth epetto fom a set of elemets s exactly the same as the umbe of ways to aage stas ad ( ) bas a ow; that s, ( + )!. Howeve, sce the stas ad bas ae dstgushable, we must dvde by both! ad ( )! to avod ovecoutg. Theoem. The umbe of -combatos of a set wth elemets, wth epetto, s ( + )!.!( )! Poof. Let A {a,a,...,a }. Cosde a stg of ( + ) bla spaces... {z } + tmes

4 Lectue, Wte 9 Page ad a set cotag Fs ad ( ) s. Each aagemet of Fs ad s to the bla spaces costtutes a - combato wth epetto, wth the umbe of Fs betwee the stat of the stg ad the fst coutg the umbe of selectos of a, the umbe of Fsbetweethefst ad secod coutg the umbe of selectos of a, ad so o. Thee s a total of C(+, ) ways to place the ( ) s to the bla spaces, ad fom ths we foce placemet of the Fs. Thus, thee s a total of C( +, ) C( +,) possble -combatos of a set wth elemets whee epetto s allowed. (+ )!!( )! Retug to ou couse eolmet example, we see that f the studet had o costats o the couses they wated to tae, the they would have a total of C( +, ) C(, ) C(, ) couse combatos to choose fom. We ca use -combatos wth epetto to calculate may othe teestg thgs; cosde, fo example, a compute algeba system that eeds to fd solutos to a gve equato. The system could aïvely chec evey possble soluto, but ths ca be slow. Usg combatos, ceta posbltes ca be uled out based o costats o othe codtos. Example. Let x, y, ad z be atual umbes. How may solutos exst fo the equato x+y+z? We ca fame ths poblem as a combatoal poblem, sce ay soluto to the gve equato coespods to a selecto of elemets fom a set of sze whee we have x elemets fom class, y elemets fom class, ad z elemets fom class. Dawg a stas ad bas dagam, we have the followg sceao: {z} x y z Sce we ae calculatg the umbe of -combatos of a set wth elemets, we get that the total umbe of solutos to the equato s C( +, ) C(8, ) C(8, ). Example. Let x, y, ad z be atual umbes, ths tme wth the costats that x, y, ad z. How may solutos exst fo the equato x + y + z? Ths example s vey smla to the pevous example, but wth added costats. Thus, we ca follow the same pocedue as befoe whle eepg md that each of x, y, ad z must tae o ceta values; amely, we must have at least two elemets fom class, at least fve elemets fom class, ad at least thee elemets fom class. Dawg a stas ad bas dagam, we have the followg sceao: FF FFFFF FFF {z} {z } {z } x y z Sce te stas ae peassged to the dagam, we must place the emag sx stas ouselves. Ths s equvalet to us calculatg the umbe of -combatos of a set wth elemets, whch leads us to coclude that the total umbe of costaed solutos to the equato s C( +, ) C(8, ) C(8, ) 8. Bomal Theoem Let s ow tae a bef step bac fom coutg ad loo at a algebac poblem: expadg bomals. As you lely leaed you fst-yea math classes (o eve eale), we ca use the FOIL method fst, oute, e, last to expad the bomal (x + y). Ths gves us the followg esult: (x + y) (x + y)(x + y) x + xy + xy + y x +xy + y. Ca we follow a smla techque fo bomals wth lage expoets? Of couse; the FOIL method s just a specal case of the dstbutve popety of multplcato, whch tells us that a(b + c) (ab + ac) fo values

5 Lectue, Wte 9 Page a, b, ad c. To see how ths wos, let s cosde (x + y) : (x + y) (x + y)(x + y)(x + y) (x +xy + y )(x + y) x + x y +x y +xy + xy + y x +x y +xy + y. Just fo fu, let s also cosde (x + y) whle we e at t: (x + y) (x + y)(x + y)(x + y)(x + y) (x +x y +xy + y )(x + y) x + x y +x y +x y +x y +xy + xy + y x +x y +x y +xy + y. By ths pot, you mght otce a ceta patte s developg. Each tem the expaso of the bomal (x + y) s of the fom ax b y c,wheeb + c ad whee a s some coe cet. How ca we calculate the value of the coe cet a fo some abtay tem wthout wtg the ete expaso? Let s beg by detemg what ths value a s coutg. We ca wte the geeal bomal (x + y) as (x + y) (x + y)(x + y) (x + y) {z } tmes By the dstbutvty popety of multplcato that we metoed eale, the expaso of ths bomal wll cota oe tem fo each possble choce of x ad y, ad we tae choces. As a llustato, assume we ae oly choosg x. If we tae x a total of tmes, the we wll add the tem x to ou expaso. O the othe had, f we tae x a total of tmes, the we must tae y a total of tmes ode to collect tems oveall. Thus, we wll add the tem x y to ou expaso. Geealzg ths dea to us choosg x a total of b tmes ad y a total of c tmes, whee b + c, wesee that the dea s equvalet to us calculatg the umbe of ways we ca choose b occueces of x fom bomals (equvaletly, choosg c occueces of y). I othe wods, we e tag a b-combato fom a set of bomals of sze (equvaletly, a c-combato), ad theefoe, the value a s equal to C(, b) C(, c). Befoe we cotue, we wll toduce a ew otato used specfcally the cotext of bomals. We say that the bomal coe cet s the umbe of ways to choose elemets fom a -elemet set, whee apple apple. Soud famla? It should; the bomal coe cet s exactly the same as a -combato, but wtte usg a d eet otato. Defto (Bomal coe cet). The bomal coe cet apple apple as C(, )!( )!., ead as choose, s defed fo Now that we ae famla wth bomal coe cets, we may use ths otato to obta the geeal fom of a bomal expaso. We obta the geeal fom by way of the bomal theoem. Theoem (Bomal theoem). Let x ad y be vaables, ad let be a atual umbe. The (x + y) X x + x y x y + x y + + xy + y.

6 Lectue, Wte 9 Page Poof. We pove by ducto. Let P () be the statemet (x + y) P Whe, we have (x + y) x + y x + y P x Assume that P () s tue fo some N. That s, assume that (x + y) P x y. y. Theefoe, P () s tue. x y. We ow show that P ( + ) s tue. Multply each sde of the equato by (x + y) to get (x + y) + x Theefoe, P ( + ) s tue. X x! y (x + y) X x! y + y X x + y + x + + x + + x X + x + + X y + + y + + y x + y. X! x y x y + X X X y + + x + y + x + y + + X x y + X x y + x + X + x + By the pcple of mathematcal ducto, P () s tue fo all N. Rema. It s possble to geealze bomal coe cets ad the bomal theoem to polyomals wth moe tha two tems, such as (x + y + z). These geealzatos ae called multomal coe cets ad the multomal theoem, espectvely. As a execse, th about how to fomulate these geealzatos. Immedately fom the statemet of the bomal theoem, we get a vaat of the theoem as a coollay. Coollay. Let x be a vaable ad let be a atual umbe. The (x + ) Poof. Follows fom the bomal theoem whe y. X x. I the poof of the bomal theoem, we eque a patcula detty that tells us somethg about the value of a bomal coe cet tems of smalle bomal coe cets. Usg ths detty, whch was amed afte the Fech mathematca Blase Pascal, we ae able to defe bomal coe cets ecusvely, whch s a geat help computatoal applcatos. Theoem 7 (Pascal s detty). Fo all apple apple, + +. y y

7 Lectue, Wte 9 Page Poof. By the defto of the bomal coe cet, we have +!( )! + ( )!( + )! (( + ) + )!( + )! ( + )!!( + )! + We ca easo about Pascal s detty the followg way. Suppose S s a set cotag +elemets, ad deote oe specal elemet as a. Let T be the subset of S ot cotag a. Thee ae + sze- subsets of S, ad these subsets ethe () do ot cota a, but oly cota elemets fom T, o () cota both a ad elemets fom T. I sceao (), thee ae possble subsets of T, ad sceao (), thee ae possble subsets of T, so altogethe we have + sze- subsets of S. Wth Pascal s detty, we ca daw a beautful tagula aagemet of bomal coe cets whee the + th tem ow +,, s deved fom the sum of the two tems ad wtte dectly above t. (Bla o oexstet etes ae tae to be zeo.) Ths aagemet s called Pascal s tagle, spte of the fact that t had bee studed cetues befoe by othe mathematcas. The fst few ows of Pascal s tagle ae as follows:. Asde fom loog ce, Pascal s tagle eveals may hdde tcaces the stuctue of ad elatoshps betwee bomal coe cets. We peset a few of these teestg esults hee wthout poof. Poposto 8. The followg dettes hold:.. P. P. P fo all apple apple (ow symmety); fo all (ow sum); fo all (ow sum of squaes); c + c+ fo all, c (colum sum). All thgs cosdeed, what do bomal coe cets ad the bomal theoem have to do wth computg? Fo oe, just as we saw ou dscusso o combatos wth epetto, the bomal theoem ca speed up ceta calculatos compute algeba systems. Multplcato s a tesve opeato fo a compute, ad may multplcatos at oce ca slow dow a computato cosdeably. Wth the bomal theoem, we ca expad ceta expessos o calculate ceta tems wth a expesso much moe e cetly.