8. Combinatorial Structures

Transcription

1 Virtual Laboratories > 0. Foudatios > Combiatorial Structures The purpose of this sectio is to study several combiatorial structures that are of basic importace i probability. Permutatios Cosider a set D with elemets. A permutatio of legth k from D is a ordered sequece of k distict elemets of D: ( x 1, x 2,..., x k ), x i D for each i ad x i x j for i j Of course, k caot be larger tha. Statistically, a permutatio of legth k from D correspods to a ordered sample of size k chose without replacemet from the populatio D. Derivatio 1. Use the multiplicatio priciple to show that the umber of permutatios of legth k from a elemet set is ( k) = ( 1) ( k + 1) 2. Show that the umber of permutatios of legth from the elemet set D (these are called simply permutatios of D) is! = () = ( 1) 1 3. Show that ( k)! = ( k)! Note that the basic permutatio formula is defied for every real umber ad oegative iteger k. This extesio is sometimes referred to as the geeralized permutatio formul Actually, we will sometimes eed a eve more geeral formula of this type (particularly i the sectio o Pólya's ur ad the beta-beroulli process). For a R, s R, ad j N, defie 4. Note that a (s, j) = a (a + s) (a + 2 s) (a + ( j 1) s) a (0, j) = a j a ( 1, j) = a ( j), a (1, j) = a (a + 1) (a + j 1), 1 (1, j) = j!. Combiatios

2 Cosider agai a set D with elemets. A combiatioof size k from D is a (uordered) subset of k distict elemets of D: {x 1, x 2,..., x k }, x i D for each i ad x i x j for i j Agai, k caot be larger tha. Statistically, a combiatio of size k from D correspods to a uordered sample of size k chose without replacemet from the populatio D. Note that for each combiatio of size k from D, there are k! distict orderigs of the elemets of that combiatio. Each of these is a permutatio of legth k from D. Derivatio The umber of combiatios of size k from a -elemet set is deoted by ( k) or C(, k). 5. Use the multiplicatio priciple ad to show that ( k ) = (k) k! A algorithm for geeratig all permutatios of size k from D is to first select a combiatio of size k ad the to select a orderig of the elemets. Thus, argue that ( k) = ( k) k!. The umber ( k) is called a biomial coefficiet. Note that the formula makes sese for ay real umber ad oegative iteger k, sice this is true of the geeralized permutatio formula ( k). With this extesio, ( k) is called the geeralized biomial coefficiet. Note that if ad k are positive itegers ad k > the ( k) = 0. By covetio, we will also defie ( k) = 0 if k < 0. This covetio sometimes simplifies formulas. 6. Show that if ad k are oegative itegers ad k the ( k ) =! k! ( k)! Basic Properties For some of the idetities i the exercises below, you are asked to give two proofs. A algebraic proof, of course, should be based o the first combiatio formula or the secod combiatio formul A combiatorial proof is costructed by showig that the left ad right sides of the idetity are two differet ways of coutig the same collectio. 7. Show that ( 0) = ( ) = 1 8. Give algebraic ad combiatorial proofs of the idetity ( k ) = ( k ) For the combiatorial argumet, ote that if you select a subset of size k from a set of size, the you leave a subset of size k behi

3 9. Give algebraic ad combiatorial proofs of the followig idetity: if ad k are o-egative itegers ad k the ( k ) = 1 ( k 1) + 1 ( k ) For the combiatorial argumet, fix a elemet of the set. Cout the umber of subsets of size k that cotai the desigated elemet ad the umber of subsets of size k that do ot cotai the desigated elemet. If each peg i the Galto board is replaced by the correspodig biomial coefficiet, the resultig table of umbers is kow as Pascal's triagle, amed for Blaise Pascal. By the Exercise 9, each iterior umber i Pascal's triagle is the sum of the two umbers directly above it. 10. Give algebraic ad combiatorial proofs of the biomial theorem: if a ad b are real umbers ad is a positive iteger, the (a + b) = k =0 ( k ) ak b k 11. Give algebraic ad combiatorial proofs of the followig idetity: if ad k are positive itegers the k ( k ) = 1 ( k 1) For the combiatorial argumet, cosider two procedures for selectig a committee of size k from a group of persos, with oe distiguished member of the committee as chair. Select the committee from the populatio ad the select a member of the committee to act as chair. Select the chair of the committee from the populatio ad the select k 1 other committee members from the remaiig 1 members of the populatio. 12. Give algebraic ad combiatorial proofs of the followig idetity: if m,, ad k are oegative itegers, the k m j =0 ( j ) ( k j) = + m ( k ) For the combiatorial argumet, suppose that a committee of size k is chose form a group of + m persos, cosistig of wome ad m me. Cout the umber of committees with j me ad k j wome, ad the sum over j. 13. Give algebraic ad combiatorial proofs of the followig idetity: if ad m are oegative itegers ad m the m j = j ( ) = m + 1 ( + 1 ) For the combiatorial argumet, suppose that we pick a subset of size + 1 from the set {1, 2,..., m}. For j {, + 1,..., m}, cout the umber of subsets i which the largest elemet is j + 1 ad sum over j. For a eve more geeral versio of this idetity, see the sectio o Order Statistics i the chapter o Fiite Samplig Models.

4 14. Show that the idetity i Exercise 9 is a special case of the idetity i the Exercise 13, as is the followig idetity for the sum of the first m positive itegers: m j =1 j = ( m ) = (m + 1) m Show that there is a oe-to-oe correspodece betwee each pair of the followig collectios. Hece the umber objects i each of these collectio is ( k). Subsets of size k from a set of elemets. Bit strigs of legth with exactly k 1's. Paths i the Galto board from (0, 0) to (, k). 16. Show that if ad k are oegative itegers the I particular, ote that ( 1 k ) = ( 1)k ( k ) = ( + k 1 1)k ( k ) Samples The experimet of drawig a sample from a populatio is basic ad importat. There are two essetial attributes of samples: whether or ot order is importat, ad whether or ot a sampled object is replaced i the populatio before the ext draw. Suppose ow that the populatio D cotais objects ad we are iterested i drawig a sample of k objects. Let's review what we kow so far: If order is importat ad sampled objects are replaced, the the samples are just elemets of the product set D k. Hece, the umber of samples is k. If order is importat ad sample objects are ot replaced, the the samples are just permutatios of size k chose from D. Hece the umber of samples is ( k). If order is ot importat ad sample objects are ot replaced, the the samples are just combiatios of size k chose from D. Hece the umber of samples is ( k). Thus, we have oe case left to cosider. Uordered Samples With Replacemet 17. Show that there is a oe-to-oe correspodece betwee each pair of the followig collectios: Uordered samples of size k chose with replacemet from a populatio D of elemets. Distiguishable strigs of legth + k 1 from a two-letter alphabet (say {*, /}) where * occurs k times ad / occurs 1 times. Noegative iteger solutios of x 1 + x x k = k.

5 18. Show that each of the collectios i Exercise 17 has +k 1 k elemets. Summary of Samplig Formulas The followig table summarizes the formulas for the umber of samples of size k chose from a populatio of elemets, based o the criteria of order ad replacemet. Samplig formulas With Order Without Order With Replacemet k +k 1 k Without Replacemet ( k) ( k) Multiomial Coefficiets Partitios of a Set Recall that the biomial coefficiet ( j ) is the umber of subsets of size j from a set S of elemets. Note also that whe we select a subset A of size j from S we effectively partitio S ito two disjoit subsets of sizes j ad j, amely A ad A c. A atural geeralizatio is to partitio S ito a uio of k distict, pairwise disjoit subsets (S 1, S 2,..., S k ) where #(S i ) = i for each i {1, 2,..., k}. Of course we must have k =. 19. Use the multiplicatio rule to show that the umber of such partitios is 1 ( 1 ) ( 2 ) ( k 1 ) ( k ) =! 1! 2! k! This umber is called the multiomial coefficiet ad is deoted by ( 1, 2,, k ) 20. Give a algebraic ad a combiatorial argumet for the idetity Sequeces ( k, k ) = ( k ) Cosider ow the set T = {1, 2,..., k}. Elemets of this set are sequeces of legth i which each coordiate is oe of k values. Thus, these sequeces geeralize the bit strigs of legth i the last sectio. Agai, let ( 1, 2,..., k ) be k a sequece of oegative itegers with i i =. =1 21. Costruct a oe-to-oe correspodece betwee the followig collectios:

6 Partitios of S ito pairwise disjoit subsets (S 1, S 2,..., S k ) where #(S i ) = i for each i {1, 2,..., k}. Sequeces i {1, 2,..., k} i which i occurs i times for each i {1, 2,..., k}. It follows that the umber of sequeces i the secod part of the Exercise 21 is ( 1, 2,, k ) Permutatios with Idistiguishable Objects 22. Suppose ow that we have object of k differet types, with i elemets of type i for each i {1, 2,..., k}. Moreover, objects of a give type are cosidered idetical. Costruct a oe-to-oe correspodece betwee the followig collectios: Sequeces i {1, 2,..., k} i which i occurs i times for each i {1, 2,..., k}. Distiguishable permutatios of the objects. It follows that the umber of permutatios i the secod part of the Exercise 22 is ( 1, 2,..., k ) The Multiomial Theorem 23. Give a combiatorial proof of the multiomial theorem. Or course, this theorem is the reaso for the ame of the coefficiets. 24. Show that there are Computatioal Exercises ( x 1 + x x k ) = k = ( 1, 2,..., k ) x 1 1 x 2 2 xk k +k 1 terms i the multiomial expasio i the Exercise 23. k I a race with 10 horses, the first, secod, ad third place fiishers are ote How may outcomes are there? 26. A licese tag cosists of 2 letters ad 5 digits. Fid the umber of tags with the letters ad digits are all differet. Arragemets 27. Eight persos, cosistig of four married couples, are to be seated i a row of eight chairs. How may seatig arragemets are there i each of the followig cases:

7 There are o other restrictios. The me must sit together ad the wome must sit together. The me must sit together. The spouses i each married couple must sit together. 28. Suppose that people are to be seated at a roud table. Show that there are ( 1)! distict seatig arragemets. The mathematical sigificace of a roud table is that there is o dedicated first chair. 29. Twelve books, cosistig of 5 math books, 4 sciece books, ad 3 history books are arraged o a bookshelf. Fid the umber of arragemets i each of the followig cases: There are o restrictios. The books of each type must be together. The math books must be together. 30. Fid the umber of distict arragemets of the letters i each of the followig words: e. statistics probability mississippi teessee alabama 31. A child has 12 blocks; 5 are red, 4 are gree, ad 3 are blue. I how may ways ca the blocks be arraged i a lie (blocks of a give color are cosidered idetical)? Committees 32. A club has 20 members; 12 are wome ad 8 are me. A committee of 6 members is to be chose. Fid the umber of differet committees i each of the followig cases: There are o other restrictios. The committee must have 4 wome ad 2 me. The committee must have at least 2 wome ad at least 2 me. 33. Suppose that a club with 20 members plas to form 3 distict committees with 6, 5, ad 4 members, respectively. I how may ways ca this be doe. Hit: the members ot o a committee also form oe of the sets i the partitio.

8 Cards A stadard card deck ca be modeled by the product set D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k} {,,, } where the first coordiate ecodes the deomiatio or kid (ace, 2-10, jack, quee, kig) ad where the secod coordiate ecodes the suit (clubs, diamods, hearts, spades). Sometimes we represet a card as a strig rather tha a ordered pair (for example q ). 34. A poker had cosists of 5 cards dealt without replacemet ad without regard to order from a deck of 52 cards. Fid the umber of poker hads i each of the followig cases: There are o restrictios. The had is a full house (3 cards of oe kid ad 2 of aother kid). The had has 4 of a ki The cards are all i the same suit (so the had is a flush or a straight flush). The game of poker is studied i detail i the chapter o Games of Chace. 35. A bridge had cosists of 13 cards dealt without replacemet ad without regard to order from a deck of 52 cards. Fid the umber of bridge hads i each of the followig cases: There are o restrictios. The had has exactly 4 spades. The had has exactly 4 spades ad 3 hearts. The had has exactly 4 spades, 3 hearts, ad 2 diamods. 36. A had of cards that has o cards i a particular suit is said to be void i that suit. Use the iclusio-exclusio formula to fid each of the followig: The umber of poker hads that are void i at least oe suit. The umber of bridge hads that are void i at least oe suit. 37. A bridge had that has o hoor cards (cards of deomiatio 10, jack, quee, kig, or ace) is said to be a Yarborough, i hoor of the Secod Earl of Yarborough. Fid the umber of Yarboroughs. 38. A bridge deal cosists of dealig 13 cards (a bridge had) to each of 4 distict players (geerically referred to as orth, south, east, ad west) from a stadard deck of 52 cards. Show that the umber of bridge deals is E28 This staggerig umber is about the same order of magitude as the umber of atoms i your body, ad is oe of the

9 reasos that bridge is a rich ad iterestig game. 39. Show that the umber of permutatios of the cards i a stadard deck is 52! E67 This umber is eormous. Ideed if you perform the experimet of dealig all 52 cards from a well-shuffled deck, you may will geerate a patter of cards that has ever bee geerated before, thereby esurig your immortality. Actually, this experimet shows that, i a sese, rare evets ca be very commo. By the way, Persi Diacois has show that it takes about seve stadard riffle shuffles to thoroughly radomize a deck of cards. Dice 40. Suppose that 5 distict, stadard dice are rolled ad the sequece of scores recorde Fid the umber of sequeces. Fid the umber of sequeces with the scores all differet. 41. Suppose that 5 idetical, stadard dice are rolle How may outcomes are there? Polyomial Coefficiets 42. Fid the coefficiet of x 3 y 4 i (2 x 4 y) Fid the coefficiet of x 5 i (2 + 3 x) Fid the coefficiet of x 3 y 7 z 5 i ( x + y + z) Geerate Pascal's triagle up to = Suppose that i a group of people, each perso shakes hads with every other perso. Show that there are ( 2) differet hadshakes. 47. I the (, k) lottery, k umbers are chose without replacemet from the set of itegers from 1 to (where k < of course). Order does ot matter. Fid the umber of outcomes i the geeral (, k) lottery. Explicitly compute the umber of outcomes i the (44, 6) lottery (a commo format).

10 For more o this topic, see the sectio o Lotteries i the chapter o Games of Chace. 48. I the Galto board game, Move the ball from (0, 0) to (12, 7) alog a path of your choice. Note the correspodig bit strig ad subset. Geerate the bit strig Note the correspodig subset ad path. Geerate the subset {1, 4, 5, 10, 12, 15}. Note the correspodig bit strig ad path. Geerate all paths from (0, 0) to (5, 3). How may paths are there? 49. A fair coi is tossed 10 times ad the outcome is recorded as a bit strig (where 1 deotes heads ad 0 tails). Fid the umber of outcomes with exactly 4 heads. Fid the umber of outcomes with at least 8 heads. 50. A shipmet cotais 12 good ad 8 defective items. A sample of 5 items is selecte Fid the umber of samples that cotai exactly 3 good items. 51. Suppose that 20 idetical cadies are distributed to 4 childre. Fid the umber of distributios are there i each of the followig cases: There are o restrictios. Each child must get at least oe cady. 52. Fid the umber of iteger solutios of x 1 + x 2 + x 3 = 10 i each of the followig cases: x i 0 for each i. x i > 0 for each i. 53. Explicitly compute each formula i the samplig table above whe = 10 ad k = Compute each of the followig: ( 5) (3) ( 1 (4) 2) ( 1 (5) 3)

11 55. Compute each of the followig: 1 2 ( 3 ) ( 5 4 ) ( ) 56. Suppose that persos are selected ad their birthdays ote Fid the umber of outcomes. Fid the umber of outcomes with distict birthdays. 57. Fid the umber of ways of placig 8 rooks o a chessboard so that o rook ca capture aother i each of the followig cases. Note that the squares of a chessboard are distict, ad i fact are ofte idetified with the product set {a, b, c, d, e, f, g, h} {1, 2, 3, 4, 5, 6, 7, 8}. The rooks are distiguishable. The rooks are idistiguishable. 58. I the sog The Twelve Days of Christmas, fid the umber of gifts give to the siger by her true love. (Note that the siger starts afresh with gifts each day, so that for example, the true love gets a ew partridge i a pear tree each of the 12 days.) 59. Suppose that 10 kids are divided ito two teams of 5 each for a game of basketball. I how may ways ca this be doe i each of the followig cases: The teams are distiguishable (for example, oe team is labeled Alabama ad the other team is labeled Aubur ). The teams are ot distiguishable. Virtual Laboratories > 0. Foudatios > Cotets Applets Data Sets Biographies Exteral Resources Keywords Feedback