= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
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1 0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet subsets of the set { A, B, C, D, E ca you fid Recall that the diffeece betwee a wod ad a set is the odeig Wheeve the ode of elemets of a set itechages, that yields to the same set That meas by chagig the ode, oe ca ot obtai a ew set O the cotay, if you chage the ode of lettes i a wod, you obtai a ew wod povided that the lettes ae diffeet Hece we eed to be caeful i computatios of diffeet collectios sice the secod expeimet above igoes those obtaied by itechagig elemets We aleady asweed the fist questio befoe Thee ae = 60 possible wods How abut the secod questio As we metioed, we eed to dop those subsets which ae obtaied by just eodeig of the same elemets Note that wheeve we have a collectio of 3 elemets, we have 3! eodeig Hece each collectio epeats itself 3! times That s why we eed to divide the total umbe of collectios by this umbe epetitios That is, ! Next, we ote that the above expessio ca be witte i a ice fom ! = ! = 5! 3!! = 5! 3! (5 3)! I geeal, the umbe of diffeet goups of items out of items (whe the ode is igoed) is give by!! ( )! We ll call this as the umbe of -combiatios of objects, ad deote it by! = k! ( )! fo 0 apple apple (Read as " choose ") R 0! = R = Example 7 A committee of 3 people will be fomed fom a goup of 0 people How may diffeet committees ca be fomed I this example, we ote that thee is o impotace of the ode of selectio If you choose peso fist, the peso the secod ad peso 3 the thid, the this selectio is the same as the selectio peso, peso, peso 3 Eve if you chage the ode of selectio, you always choose the same 3 peso Hece they all fom the same goup That s why this is a questio about umbe of combiatio of 3-people out of 0 As we discussed above, that umbe equals 0 3 = 0! = = 40 3! 7! 3
2 4 Combiatios Copyight by Deiz Kalı Example 8 Fom a goup of 5 wome ad 7 me, i how may diffeet committees cosistig of wome ad 3 me ca be fomed As i the pevious example, this is a combiatio poblem Hece we eed to coside expeimets; choosig wome out of 5, ad 3 me out of 7: 5 7 = ii R what if of the me efuse to be i the same committee We ca appoach this poblem i ways st way: Let s say these me ae called M ad M How may of these committees do iclude these me Ay wome- M - M - Ay me except M ad M " 5 " 5 Hece 5 5 = 50 of these committees iclude both M ad M Sice we wat to kow the umbe of those which do t iclude both, we subtact 50 out of the total umbe of goups, that is, = 300 d way: We coside 3 possible cases: Case : Committees without ay of these me Case : Committees with M but without M Case 3: Committees with M but without M Whe we cout the umbe of committees i each case, we obtai 5 5 Case : Case : 5 5 Case 3: ad by addig these umbes togethe, we have the total umbe = It is a good time to ote a useful idetity o combiatios If ad ae positive iteges ad apple apple the we ca wite = + This is a easy expessio to emembe Note that the left had side of the equatio epesets the umbe of diffeet ways to choose elemets out of objects This ca be doe i the followig way: Fist fix oe of the objects
3 Combiatoial Aalysis {z fixed object emaiig (-) objects The the -goups fomed out of objects eithe cotais this fixed object -goup icludig the fixed oe ad (-) chose fom (-) emaiig objects o does ot cotai this fixed oe Copyight by Deiz Kalı -goup excludig the fixed oe, hece chose fom (-) The umbe of -goups cotaiig this fixed object is emaiig objects icludig the fixed oe sice i this case you eed to choose the emaiig objects out of Similaly, the umbe of -goups, ot cotaiig the fixed object, is, sice you eed to choose all of -elemets fom the emaiig objects So the equality atually appeas Biomial Theoem We begi with a questio: Questio 5 How do you expad (x + y) o (x + y) 3 Let s expad (x + y) fist (x + y) =(x + y)(x + y)=xx + xy + yx + yy = x + xy + y Note that the commutativity of multiplicatio allows us to wite yx = xy ad hece we have of the tem xy i the expasio Next, (x+y) 3 =(x+y)(x+y)(x+y)=xxx+xxy+xyx+yxx+xyy+yxy+yyx+yyy = x 3 +3 x y+3 xy + y 3 Similaly, commutativity leads to xxy = xyx = yxx ad yyx = yxy = xyy To detemie the coefficiet of x y, fo example, it is eough to cout diffeet ways i which oe ca place the symbol x twice i 3 possible seats x x x x x x Oce all x s ae places, the emaiig seats ae eseved fo the othe symbol y Hece the umbe of x y tems is the umbe of diffeet ways of choosig seats out of 3, which is 3 I a simila way, we ca cout the umbe of xy All we eed to do is to place the oly oe symbol of x i oe of 3 seats
4 4 Combiatios 3 x x x Clealy, this ca be doe i 3 ways So we have idea of detemiig the coefficiets i the expasio The (x + y) 3 = x 3 + x y + xy + y We ca geealise this idea to ay biomial expasio Copyight by Deiz Kalı Theoem 4 Biomial Teoem If is a positive itege the (x + y) = Â x y Example 9 What is the coefficiet of x 7 y 4 i the expasio of (x + y) As we deduced above, we eed to choose 7 seats fo x out of possible seats That is, 7 Note that we ca also choose seats fo y istead of x I this case, the total umbe will be 4 which is the same as the pevious esult =0 Example 0 How may -elemet subsets does a -elemet set have This questio is vey simila to the oe above To fom a -elemet subset, we eed to choose elemets fom the mai set Moeove, it etus the same set if you choose the same elemets i a diffeet ode Hece total umbe of -elemet sets is Now let s elate the above example to the Biomial Theoem Example How may subsets does a -elemet set have We basically cout each possible subset " " " " empty set -elemet subsets -elemet subsets -elemet subsets Note that this is the sum of coefficiet i Biomial Theoem, which ca be obtaied by settig x = ad y = Hece = 0 Â =( + ) = =0
5 4 Combiatoial Aalysis Questio 6 Coside the gid below: C B Copyight by Deiz Kalı A Suppose you wat to walk o this gid fom the poit A to the poit B Thee ae some ules Each step must be take fom oe coe to a adjacet coe Each step ca be take oe uit up o oe uit to the ight O how may diffeet paths ca you walk fom A to B Note that thee 4 steps up ad 9 steps to the ight to aive at the poit B These steps must be fulfilled i ay ode If you chage the odes of these steps, you obtai a diffeet path That meas the umbe of paths is basically the umbe of aagemets of these UP (") ad RIGHT(!) steps Fo example, "!"!!""!!!!!! is path To see the coectio, we ca assig x to! ad y to " The the path is yxyxxyyxxxxxx Hece the umbe of diffeet paths is just the coefficiet of x 4 y 9, which is 3 4 What if you have make a stop at the poit C The you eed to pefom two expeimets i ode Fist, walk fom A to C, which ca be made i 5 3 diffeet ways The walk fom C to A by 8 ways Hece thee ae diffeet path passig though the poit C
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