3.1 Counting Principles

Transcription

1 3.1 Coutig Priciples Goal: Cout the umber of objects i a set. Notatio: Whe S is a set, S deotes the umber of objects i the set. This is also called S s cardiality. Additio Priciple: Whe you wat to cout a set which you are able to break dow ito subsets, the cout these parts idividually ad take their sum. Mathematically, suppose that S is a set of objects. We say that S 1, S 2,..., S k is a (set) partitio of S if S = S 1 S 2 S k S i S j = for all i ad j. (*) Additio Priciple: If S 1,S 2,...,S k is a partitio of S, the S = S 1 + S S k. The additio priciple is used to break dow a larger set ito more maageable pieces. 11

2 3.1 Coutig Priciples Example: A studet wats to take either a math class or a biology class to keep his workload dow. If there are three math classes ad four biology classes to choose from, how may choices are there i all? A: Assumig there is o cross-listed course, = 7. Arrage yourselves i groups of two or three for *Partitio Boggle* Example: You are orgaizig the yogurt sectio of the store; determie how may types of yogurt: (a) if there are te flavors ad three styles. (b) if i additio there are four brads for each. (c) if i additio there are two sizes each. 12

3 3.1 Multiplicatio Priciple This is the Multiplicatio Priciple: If a first task has p outcomes ad a secod task has q outcomes for all outcomes of the first task, the the two tasks performed successively have pq outcomes. Multiplicatio Priciple Practice Groupwork: 1) How may 2-digit umbers have o-zero digits? 2) How may two-digit umbers have distict ad o-zero digits? 3) How may odd umbers betwee 1000 ad 9999 have distict digits? [Hit: It may be useful to choose the digits i a o-stadard way.] 4) How may poker hads are full houses? [Poker hads cotai five cards; a full house has three cards of oe value ad two cards of a differet value.] Aother approach to 2): 13

4 3.1 Subtractio Priciple Let A be a set ad U be a larger set cotaiig A. Defie the complemet of A i U, writte A or A c, as the objects i U ot i A. I other words, A c = U \ A. Subtractio Priciple: Let A U. The A c = U A. Example: If computer passwords cosists of the digits 0 9 ad the letters a z, the how may passwords have a repeated symbol? Total possibilities Distict-digit possibilities ,176,782,336 1,402,410,240 Total: 774,372,096 (About 35% of the total #.) Divisio Priciple: Let S be partitioed ito k parts of the same size. The k = S part. 14

5 Permutatios ad Combiatios The material i Chapter 3 cosists of how to cout arragemets of objects. Two types: A r-permutatio of a set S is a ordered arragemet of r of its elemets. A r-combiatio of a set S is a uordered arragemet of r of its elemets. Cosider the set S = {a, b, c}: r =1 r =2 r =3 r =4 r-permutatio of S r-combiatio of S Whe we discuss permutatios of a set S with o referece to a r, the we are arragig all of S s elemets. Notice: It makes o sese to discuss a combiatio of a set. 15

6 Coutig Arragemets Notatio:! =( 1)( 2) 2 1. By covetio, 0! = 1 How may r-permutatios are there of a -elemet set? P (, r) := ( 1)( 2) ( r + 1) =! ( r)!. P (3, 1)= 2! 3! =3 P (3, 2)= 3! 1! =6 P (3, 3)= 3! 0! =6 How may r-combiatios are there of a -elemet set? Notatio: C(, r) = ( ) r choose r Theorem P (, r) =r! ( ) r Corollary. I factorial otatio, ( ) r =! r!( r)! 16

7 Proof of Theorem Theorem P (, r) =r! ( ) r Proof: Let S have elemets. The r-combiatios of S ad r-permutatios of S are related i the followig way: Every r-permutatio of S ca be geerated i exactly oe way usig the followig steps: 1. Choose r elemets from S. 2. Order the r elemets i some way. There are ( ) r ways to choose r elemets from S, ad r! ways to permute these r elemets. By the multiplicatio priciple, P (, r) =r! ( ) r. Aother proof is by way of the divisio priciple: Q: How may r-permutatios of S cotai the exact same elemets? A: r!, sice r elemets ca be arraged i r! ways. If we look at all r-permutatios of S ad disregard order, the each r-combiatio of S appears r! times. Therefore, ( ) r = P (,r) r!. 17

8 Arragemet Examples Example: How may 4-letter words ca be formed from the letters {a, b, c, d, e}? Example: I how may ways ca erolled studets atted class? out of the Example: I how may ways ca these seat themselves i the chairs? studets Example: I how may ways could the istructor see studets i chairs? Example: How may seve-digit umbers are there such that the digits are distict, take from {1, 2,..., 9}, ad such that 5 ad 6 do ot appear cosecutively i either order? 18

9 3.2 Circular Permutatios Example: If six childre are marchig i a circle, how may differet ways ca they form their circle? We eed to be careful because multiple circular arragemets are equivalet: This is a example of a circular permutatio; this is i cotrast to the liear permutatios that we dealt from before. We ca use the divisio priciple to cout circular permutatios. We otice that there are liear permutatios for every circular permutatio. Theorem I geeral, the umber of circular r-permutatios of a set of elemets is: 19