The number of r element subsets of a set with n r elements

Transcription

1 Popositio: is The umbe of elemet subsets of a set with elemets Poof: Each such subset aises whe we pick a fist elemet followed by a secod elemet up to a th elemet The umbe of such choices is P But this pocess couts each subset! times, oe fo each pemutatio of the subset Example: A committee of 4 is to be chose fom a goup of 10 people I how may ways ca this be doe? If thee ae 6 me ad 4 wome i the goup, how may of the possible committees will have 2 me ad 2 wome? No wome? Suppose two of the goup efuse to seve o a committee togethe How may committees ae ow possible? 10 The fist umbe is 10987/ The secod situatio ivolves choosig 2 of the 6 me ad 2 of the fou wome The poduct piciple applies to give the umbe as If thee ae to be o wome o the committee we must choose 4 fom the 6 6 me givig 6543/ I the last pat it 4 is easie to cout the umbe of committees with the disageeable two ad apply the subtactio piciple to get 10 8 umbe Hee the umbe of committees icludig the difficult pai ae chose by pickig those 2 i 1 way ad pickig aothe 2 fom the emaiig 8 Theoem: of a + b is Example: Biomial Theoem The coefficiet of a b i the expasio Fo 6 we get a + b 6 1a 6 + 6a 5 b + 15a 4 b a 3 b a 2 b 4 + 6ab 5 + 1b 6

2 Fo example the 15 tems ivolvig a 4 b 2 ae aaaabb, aaabab, aaabba, aabaab, aababa, aabbaa, abaaab, abaaba, ababaa, abbaaa, baaaab, baaaba, baabaa, babaaa, bbaaaa, oe fo each choice of 4 places fom 6 fo the a s Poof: of Biomial Theoem Expad a + b ito 2 tems whee we keep tack of the ode the tems, that is, do ot eplace ba by ab Now collect tems The umbe cotibutig to a b is equal to the umbe of ways of pickig places fom fo the a s, ad thus is equal to 8 Example: The coefficiet of x 5 i the expasio of 1 + x 8 is Example: The coefficiet of x 5 i the expasio of 2 + x 8 is Pascal s Tiagle Pascal s Tiagle is the ame give to a aagemet of all the biomial coefficiets i a tiagula patte with the umbes ceted o the th ow i ode of iceasig fom left to ight

3 Eteig values fo the biomial coefficiets gives Popositio: The followig idetities hold + 1, + 1

4 Poof: Fo the fist idetity we ca compute diectly!!!!!! Alteatively, usig the itepetatio of as the umbe of elemet subsets of a set A of size, ote that each choice of a elemet subset B of A automatically idetifies a -elemet subset A B Fo the secod idetity we ca agai compute diectly! + 1!! +! 1! + 1! 1!!! + 1 1! + 1!! + 1 +! + 1! + 1!! + 1! + 1 Alteatively, agai usig the itepetatio of + 1 as the umbe of elemet subsets of a set A of size + 1, we ca sigle out a elemet a 1 of A ad split the set of elemet subsets of A ito those cotaiig a 1 ad those ot cotaiig a 1 Fo a subset of the secod type, we simply choose elemets fom the emaiig elemets, givig ways Fo a subset of the fist type, we use the poduct piciple to choose the elemet a 1 i 1 way ad the emaiig 1 elemets fom the emaiig elemets, givig 1 ways 1 Note: We have see that P is the umbe of pemutatios of objects take at a time, while C is the umbe of subsets of size fom The diffeece betwee P ad C is that ode is impotat i the fist case but ot i the secod We have also see that is the umbe of ways of choosig objects fom if ode is impotat ad each chose object is

5 eplaced befoe the ext choice is made This leaves oe othe case, whee we select objects fom with eplacemet but ode is ot impotat Example: Suppose 5 cads ae chose fom a stadad deck of 52 How may diffeet distibutios of heats, diamods, spades ad clubs ae possible i such a had? Thee ae 56 such diffeet distibutios We ca cout them by odeig the suits ad listig the possible ways of distibutig 5 amog the 4 suits Howeve thee is a easie way Defiitio: A -selectio fom is a uodeed selectio of objects fom with epetitio allowed Theoem: The umbe of -selectio fom is Poof: Stat by odeig the types of elemets fom 1 to Fo each - selectio aage the elemets of the selectio so that type 1 elemets appea fist, type 2 elemets appea ext, etc Betwee each type of elemet i the selectio put a sepaatig make of the fom xxx yy icludig exta makes fo types uepeseted i the selectio: xxx zz The esult is a stig of legth + 1 sice thee ae elemets ad we eed 1 makes to sepaate the types Theefoe a -selectio ca be idetified with a choice of 1 places fo the makes i a stig of legth + 1 Example: Retuig to the 5 cads chose fom a stadad deck of 52, the umbe of diffeet distibutios of heats, diamods, spades ad clubs i such a had is the umbe of 5-selectios fom 4 objects ad is thus Example: 3 dice ae thow How may distibutios of the umbes 1, 2, 3, 4, 5, 6 ae possible?

6 We ae selectig 3 uodeed thigs fom 6 with epetitio so the umbe is Note: We summaise the fomulae fo the umbe of ways of choosig objects fom, whe ode is impotat o ot ad epetitio is allowed o ot epeated odeed -samples uepeated!! -pemutatios uodeed + 1 -selectios 1 -combiatios Multiomial coefficiets: Ou fial cocept i coutig will be multiomial coefficiets We will itoduce this idea usig thee examples, which all compute the same umbe Example: NAVAN? How may five-lette wods ca be fomed usig the lettes of NNAAV, NNAVA, NNVAA, NANAV, NANVA, NAANV, NAAVN, NAVNA, NAVAN, NVNAA, NVANA, NVAAN, ANNAV, ANNVA, ANANV, ANAVN, ANVNA, ANVAN, AANNV, AANVN, AAVNN, AVNNA, AVNAN, AVANN, VNNAA, VNANA, VNAAN, VANNA, VANAN, VAANN 30 i total These ca be couted by lookig at possibilities fo fist lette, the secod, etc, Howeve it is easie to fist distiguish the two N s ad the two A s to give 5! wods ad the ote that these wods ae i goups of fou which all ead the same if the two N s ad the two A s ae udistiguished Theoem: The umbe of aagemets of objects of which 1 ae idetical, 2 ae idetical,, ae idetical is! 1! 2!! Example: I how may ways ca a goup of 5 people be patitioed ito 3 odeed sets of sizes 2, 2 ad 1?

7 Beak the poblem ito 3 stages ad use the poduct piciple Fist choose 2 fom 5 fo the fist subset, the 2 fom 3 fo the secod ad fially 1 fom 1 fo the last The total is ! 3! 1! !3! 2!1! 1!0! 5! 2!2!1! Note the cacellig of the ight had tem below the lie with the tem above the lie of the ext facto Theoem: The umbe of ways of patitioig a set of objects ito odeed subsets of sizes 1, 2,, is! 1! 2!! Example: What is the coefficiet of x 2 y 2 z i x + y + z 5? Expadig while keepig tack of the ode of lettes will yield 3 5 tems The oes cotibutig to x 2 y 2 z will have 2 x s, 2 y s ad 1 z The umbe will be the umbe of ways of patitioig 5 ito odeed subsets of sizes 2, 2 ad 1 ad is thus 5! 2!2!1! Theoem: If the the coefficiet of x 1 1 x 2 2 x x 1 + x x is! 1! 2!! Defiitio: Give positive iteges 1, 2,, ad , we defie the coespodig multiomial coefficiet to be! 1, 2,, 1! 2!! Note: Whe 2 we have the usual biomial coefficiets If 1 s < the set t s ad wite! s + t! s + t s s! s! s!t! s, t i

8 Note: A multiomial coefficiet ca be itepeted as a coefficiet i a expasio, the umbe of odeed patitios of a fiite set o the umbe of eaagemets of a stig of lettes with epetitios Example: How may diffeet ways ca a goup of 12 studets be aaged ito thee teams of 4 if a the teams ae odeed, ad b the teams ae ot odeed Fo a the umbe is 12 4, 4, 4 12! 4!4!4! 34, 650 Fo b, we ca pemute the 3 teams ad the umbe is ! 5, 775 3! 4, 4, 4 3!4!4!4! Example: I how may ways ca 52 cads be dealt out evely amog 4 people? Hee the umbe is 52 13, 13, 13, 13 52! !13!13!13! Example: How may eleve-lette wods ca be fomed usig the lettes of MISSISSIPPI? Hee the umbe is 11 1, 4, 4, 2 11! 34, 650 1!4!4!2!