1. How many possible ways are there to form five-letter words using only the letters A H? How many such words consist of five distinct letters?

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1 COMBINATORICS EXERCISES Stepha Wager 1. How may possible ways are there to form five-letter words usig oly the letters A H? How may such words cosist of five distict letters? 2. How may differet umber plates for cars ca be made if each umber plate cotais two letters (A Z) followed by five digits (0 9)? 3. We wat to desig a flag that cosists of three horizotal stripes; the colour of the middle stripe should be differet from the other two stripes. How may possibilities are there, if the colours red, gree, blue, yellow, blac ad white ca be used? 4. How may diagoals does a regular dodecago (a twelve-sided polygo) have? 5. How may three-digit umbers abc have the property that a b c? 6. How may differet four-digit umbers are there such that the product of the four digits is 420? 7. The dea of sciece wats to select a committee cosistig of mathematicias ad physicists to discuss a ew curriculum. There are 15 mathematicias ad 20 physicists at the faculty; how may possible committees of 8 members are there, if there must be more mathematicias tha physicists (but at least oe physicist) o the committee? 8. A palidrome is a word that ca be read the same way i either directio (such as RACE- CAR). How may 9-letter palidromes (ot ecessarily meaigful) ca be formed usig the letters A Z? 9. The four wome Ae, Betsie, Charlotte ad Dolores ad the six me Eric, Fra, George, Harry, Ia ad James are frieds. Each of the wome wats to marry oe of the six me. I how may ways ca this be doe? 10. How may five-elemet subsets of {1, 2, 3,..., 10} cotai at least oe odd elemet? 11. Determie the coefficiet of x 3 y 4 z i the expasio of (a) (x + y 2 + z) 6 (b) (2x y 3z) I how may possible orders ca the letters of the word MATHEMATICS be arraged? 13. At a dace, there are 20 girls ad 20 boys. How may ways are there to form 20 pairs? How may, if boys may dace with boys ad girls with girls? 14. Cosider seueces (x 1, x 2,..., x ) of legth whose elemets are tae from the set {1, 2,..., }. Determie the proportio of those seueces that do ot cotai the umber 1; what happes as? 15. How may words of legth ca be formed from the letters A,B,C if the letter A has to occur a eve umber of times? 16. Determie the umber of pairs (A, B) of sets such that A is a subset of B ad B is a subset of {1, 2,..., }.

2 17. Prove the idetity i two differet ways: m ( ) m ( ) + 1 m + 1 by iductio o, by a coutig argumet: if m + 1 umbers are chose from the set {0, 1,..., }, how may choices are there such that the largest umber chose is? 18. Fid a formula for the sum ( ) by meas of the biomial theorem. [HINT: differetiate!] What ca be deduced about the average umber of elemets i a radom subset of {1, 2,..., }? 19. Cosider all r-elemet subsets of the set {1, 2,..., }. Each of them has a maximum; prove that the sum of these maxima is ( ) 1 r 1 r ad show that this sum is eual to r ( ) ( +1 r+1. [HINT: show first that 1 ( r 1) r r), ad mae use of the previous problem.] Deduce that the average maximum of a r-elemet subset of {1, 2,..., } is precisely ( + 1). r r We wat to cout the umber of ways to choose five elemets from the set {1, 2,..., 20} with the restrictio that we may ot choose cosecutive itegers. Why is this umber eual to the umber of positive iteger solutios to the euatio l 1 + l 2 + l 3 + l 4 + l 5 + l 6 21 with the additioal restrictio that l 2, l 3, l 4, l 5 > 1? [HINT: Cosider differeces] Substitute m 2 l 2 + 1,..., m 5 l to obtai l 1 + m 2 + m 3 + m 4 + m 5 + l 6 17, where l 1, m 2, m 3, m 4, m 5, l 6 ca be arbitrary positive itegers. Use the dots-ad-bars argumet to determie the umber of solutios. I geeral, i how may ways ca elemets be chose from {1, 2,..., } if o cosecutive umbers are allowed? 21. Prove that for all > 0. ( ) ( 1) A ma has a certai umber of frieds; he wats to ivite three of them for dier every day of the year. How may frieds must he have at least if he does ot wat to ivite the same three frieds twice?

3 23. A sigle piece is placed o the lower-left corer suare of a 8 8-chessboard. The piece may oly move horizotally or vertically, oe suare at a time. How may possible ways are there to move the piece to the opposite corer i 14 moves (the smallest possible umber of moves)? 24. Prove the formula ( ) ( ) ( m + 1 m + 1 m ) ( ) + m 1 by comparig coefficiets i the idetity (1 + x) +2 (1 + x) 2 (1 + x). 25. How may 3-elemet subsets of {1, 2, 3,..., 100} cotai at least oe elemet that is divisible by 2 ad at least oe elemet that is divisible by 5? 26. Each of the ie uit suares of a 3 3-suare is coloured radomly red or blue, each with probability 1 2. Determie the probability that oe of the four 2 2-suares is completely red. 27. Accordig to a recet survey, 60% of Stellebosch studets play rugby, 50% play cricet, ad 70% play teis. Furthermore, it was foud that 30% play both rugby ad cricet, 35% play rugby ad teis, ad 30% play cricet ad teis. Someoe claims that 20% play all three sports. Show that this caot be true. 28. How may umbers betwee 1 ad are either a suare or a cube of a iteger? 29. The South Africa Natioal Assembly cosists of 400 members. How may possible ways are there to divide the 400 seats amog three parties (a) such that oe of them has a majority? (b) such that oe of them has a 2/3-majority? 30. Bridge is played with a stadard dec of 52 cards, each player receives 13 cards. How may possible hads ca a player get i a game of bridge? I the HCP (high card poits) system, four poits are assiged to a ace, three poits to a ig, two to a uee ad oe to a jac. How may possible hads result i a total of (a) exactly three poits? (b) at least three poits? 31. Prove the idetity ( ) + 1 i( i) 3 i0 by a combiatorial argumet: If three umbers are chose from the set {0, 1,..., }, how may choices are there such that the middle oe is i? Use the same idea to determie the sum i0 ( i r )( i 32. Determie a formula for the umber of permutatios of {1, 2,..., } with exactly p fixed poits. What is the probability that a radomly chose permutatio has p fixed poits, as? 33. Prove the idetity ( ) ( )( ) by meas of a appropriate polyomial idetity. l ). (( ) ( )) ( 1) 2

4 34. Solve the followig ohomogeeous recursios: (a) a 3a 1 2a ; a 0 3, a 1 6. (b) a a 1 + 2a ; a 0 0, a Let F deote the Fiboacci seuece, defied by F 0 0, F 1 1 ad F +1 F + F 1. Show that F F Suppose that A(x) is the geeratig fuctio of a seuece a. What is the geeratig fuctio of the seuece (a) 3 a, (b) a? 37. We cosider the umber of triagulatios of a regular -go (the figure shows some possible triagulatios i the case 6; there are fourtee triagulatios i this case). Show that the umber of triagulatios is a Catala umber. 38. A certai computer system oly allows passwords that obey to the followig rules: A password is a combiatio of ay of the te digits 0-9, the 52 (uppercase ad lowercase) letters ad the eight additioal characters!,$,%,&,,*,(,). Ay letter or special character has to be followed by a digit. Determie the umber of possible passwords whose legth is (a) eight (b) at most eight by meas of geeratig fuctios. 39. Cosider the followig variat of the coi stac problem: this time, we form stacs of suares; a cofiguratio cosists of rows of cotiguous suares placed o top of each other i such a way that the leftmost ad the rightmost suare i every row remai uoccupied. The figure shows a possible cofiguratio. How may stacs of this type are there, if the bottom row cosists of suares?

5 40. Determie the umber of compositios of a iteger ito odd summads by meas of geeratig fuctios ad the symbolic method (for istace, is a feasible compositio of 13). 41. Solve the followig oliear recursio by meas of geeratig fuctios: a a, a 0 0, a Gradma wats to reward her four gradchildre; she has a amout of R100 available, ad wats to distribute it accordig to the followig rules: The oldest should get at least R25, The yougest should ot get more tha R20. Determie the umber of possible ways to distribute the moey. 43. Usig the symbolic method, determie the expoetial geeratig fuctio for words over the alphabet {A,B,C} with the property that every letter occurs at least twice. 44. Motzi paths are defied lie Dyc paths, the oly exceptio beig that horizotal ( level ) steps are possible as well. The figure shows a example of a Motzi path. Determie the geeratig fuctio for Motzi paths by meas of the symbolic method. 45. Determie the geeratig fuctio for uary-biary trees, that is, rooted trees with the property that every ode has either oe or two childre (iteral ode), or o child (exteral ode). 46. Recall that a deragemet is a permutatio without fixed poits (euivaletly, a permutatio without a cycle of legth 1). Show that the expoetial geeratig fuctio for deragemets is give by e x 1 x, ad deduce the formula! ( 1) for the umber of deragemets of {1, 2,..., }. 47. A perso moves o a lie, oe step to the left (with probability p) or oe step to the right (with probability 1 p) at every secod. (a) Show that there are 2 +1( 2 ) seueces of steps that tae the perso bac to the origial positio for the first time after exactly steps ( 0).!

6 (b) Show: the probability for the perso to retur to the origial positio at some stage is give by ( ) 2 2 p (c) Write this expressio i terms of the geeratig fuctio for the Catala umbers ad simplify to prove that the probability is exactly 1 p. 48. Determie formulas for the followig special Stirlig umbers: [ ] { } (a), (b) I how may differet ways ca billiard balls umbered 1 to 8 be coloured i four colours (red, blue, gree, yellow) if every colour has to be used at least oce? 50. Every permutatio ca be broe up ito rus (cosecutive elemets i icreasig order) i a uiue way. For istace, the permutatio has four rus, idicated by the bars. How may permutatios of ie elemets have exactly four rus? 51. Determie the umber of permutatios of {1, 2,..., 7} with at least 4 cycles. 52. Let σ be a permutatio of {1, 2,..., }. For every j, we defie b j to be the umber of elemets to the left of j that are larger tha j. The seuece b 1, b 2,..., b is called the iversio table of σ. (a) Determie the iversio table of the permutatio σ (b) Determie the permutatio σ whose iversio table is 4, 4, 3, 2, 2, 0, Show that [ ] ( 1) x x x(x 1)(x 2)... (x + 1). 54. Show that [ ] j [ ]! j. j! HINT: How may permutatios of {1, 2,..., + 1} are there for which 1 is i a cycle of legth + 1 j? 55. Show that { } j ( ){ } j. j HINT: How may set partitios of {1, 2,..., + 1} are there for which 1 is i a group of + 1 j elemets?

7 56. Show that m ( ) { } m! 1 m 1 { } m m by meas of a combiatorial argumet: how may fuctios from {1, 2,..., } to {1, 2,..., m} have exactly distict values i their rage? 57. Prove: if > 1, the the umber of permutatios of {1, 2,..., } with a eve umber of cycles euals the umber of permutatios with a odd umber of cycles. 58. A permutatio ca be regarded as a bijectio from {1, 2,..., } to {1, 2,..., }. Therefore, ay permutatio has a iverse. For example, the iverse of is the permutatio (a) Why do a permutatio ad its iverse have the same umber of cycles? (b) Show: a permutatio that is eual to its ow iverse cosists oly of 1- ad 2-cycles. (c) Prove by meas of (a) ad (b): if < /2, the [ ] is eve. 59. A Joyce tree is a biary tree whose odes are placed at distict levels, see the figure. How may Joyce trees with odes are there? 60. The geeratig fuctio A(x) 1 a x satisfies the euatio A(x) x (1 A(x)) 2. Determie a explicit formula for the coefficiets a by meas of the Lagrage iversio formula. 61. Show that the expoetial geeratig fuctio of the seuece ( 1) /2 1 (4 2 )B (where B is the -th Beroulli umber) is ix + x ta x, ad deduce the formula B ( 1) / t 1 for all eve values of, where t 1 deotes a taget umber. 62. Draw the Ferrers diagram of the partitio , ad determie the cojugate partitio. 63. I how may ways ca oe give chage of rad usig R1 or R2 cois? 64. Determie geeratig fuctios for the followig types of partitios: (a) Partitios with the property that 1 occurs a eve umber of times (possibly 0) as a part, while 2 does ot occur as a part at all. (b) Partitios with the property that 1 does ot occur as a part. Show that the geeratig fuctios i (a) ad (b) are the same.

8 65. Use geeratig fuctios to show that the umber of partitios of i which every part occurs at most twice is the same as the umber of partitios of with the property that oe of the parts is divisible by Prove that the -biomial coefficiets [ ] (1 )(1 2 )... (1 ) (1 )(1 2 )... (1 ) (1 )(1 2 )... (1 ) satisfy the recursio [ ] + 1 [ ] [ ] Prove the -biomial theorem (Cauchy biomial theorem): (1 + j x) j1 [ ] x (+1)/2. The special case 1 is the ordiary biomial theorem. 68. A partitio that is eual to its cojugate is called a self-cojugate partitio. Show that the umber of self-cojugate partitios of is the same as the umber of partitios of ito distict odd parts. [HINT: loo at the figure]