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1 Books Recommeded for Further Readig by o 0//8. For persoal use oly.. K. P. Bogart, Itroductory Combiatorics rd ed., S. I. Harcourt Brace College Publishers, R. A. Brualdi, Itroductory Combiatorics 5th ed., Pretice Hall, C. C. Che ad K. M. Koh, Priciples ad Techiques i Combiatorics, World Scietific, 99.. D. I. A. Cohe, Basic Techiques of Combiatorial Theory, Joh Wiley & Sos, R. L. Graham, D. E. Kuth ad O. Patashik, Cocrete Mathematics d ed., Addiso-Wesley, 99.. B. W. Jackso ad T. Dmitri, Applied Combiatorics with Problem Solvig, Addiso-Wesley, C. L. Liu, Itroductio to Combiatorial Mathematics, McGraw-Hill, F. Roberts ad B. Tesma, Applied Combiatorics d ed., Pretice Hall, A. Tucker, Applied Combiatorics th ed., Joh Wiley & Sos,

2 by o 0//8. For persoal use oly. This page itetioally left blak

3 Aswers to Exercises by o 0//8. For persoal use oly..., 5,, 55, r= r. i 0 ii i ii 5. i 0 ii m iii mt. 0.5 i ii iii iv. i 7 ii 05 iii 78.!. v 9! vi 8! vii 9!! viii 7!5! ix 5!! x 8!9 8 7 xi 5!5! xii iii 78 iv 80 v i 0! ii 8!! iii 7! 8 7 iv 7!! i 5! ii!! iii!.7 i 7 ii 7 7 iii

4 00 Coutig by o 0//8. For persoal use oly..9 i 0 ii P 0 iii i 9 ii 9. i ii 7. 7 i 7 7 ii 7 7. a 5 0 b , 0 5. a i 0 ii b By FTA, express as = p m pm p m k k. The umber of positive divisors is k i= m i , 5 7, 5 7, 5 7, 5 7, i 5 7 ii 5 iii 5 iv i ii 5 5 iii iv ! 5. 0!!.! 7. i 55 v 7,0 7. i i 7 7. i ii 7 ii 5 iii 55 ii 9 7 ii r r 7.8 r iii iv

5 Aswers to Exercises 0 by o 0//8. For persoal use oly. 7.!7 7. i!5! ii! iii! 5 iv 5!5! 7. i 0! ii!!0 7.5 i!! ii! 8. i 9 ii P 9 8. i 8 ii m i! ii 9. 0! 5!! 9. 9!!!,!! 9. i 5!!5!! ii 0 5 iii 9. i m ii Pm iii 9.5 i! ii 9. 0!! iii 5 9 m iv m! iii! 9.7!,!,!! 0. 5 x, 0. k =;x ; 0. 7 x 7, 8 x !!.7 i k k= ii.8 k =, = i 7 ii.5 i 7 ii 80. i 87 ii 7 iii x 5, x!!!

6 0 Coutig by o 0//8. For persoal use oly Hit: Fori =,,...,,leta i be the set of ways such that Couple i are seated together.. 0 Hit: Divide ito cases accordig to the umber of matchigs ad fid the correspodig umbers of deragemets.. Hit: Fori =,,...,, let A i be the set of iteger solutios with x i 0.. Hit: Fori =,,...,0, let A i be the set of ways such that Lady i gets back her hat ad B i be the set of ways such that Lady i gets back her umbrella..5 Hit: Fori =,,...,,leta i be the set of mappigs where i is ot a image.. ia 5 8 b 5 7 c 0505 ii Hit: LetA i be the set of ways without jersey i, fori =,,,, 5. iii The umber of ways the team ca choose a jersey from 5jerseysfor matches if the team uses each jersey at least oce; 0 5. Hit: Objects sums alog the rows, colums ad diagoals; Boxes all possible sums. 5. Hit: Objects coordiates of 5 lattice poits; Boxes all permutatios of parities odd or eve of the x ad y coordiates. 5. Hit: Objects 9 poits; Boxes i squares of side uit withi bigger square; ii 9 squares of side uits withi bigger square. 5.. Hit: Follow the proof of Example 5.5. Show also a sequece of distict umbers where there is o icreasig or decreasig subsequeces of k umbers,for k =. 5.5 Hit: Suppose oe of the boxes cotai at least m objects. 5. Hit: Suppose for all i =,,...,,theith box cotais less tha k i objects. 5.7 Hit: Objects pairs formed from the objects; Boxes possible absolute differeces. Watch out for a twist!

7 Aswers to Exercises 0 by o 0//8. For persoal use oly. 5.8 Hit: Objects the kigs; Boxes squares of side uits. 5.9 PP is wrogly used.. i b =.05b X for, b = 0000 X, b = X. ii X = to the earest iteger.. s =s s for, s =,s = 5.. s = s for, s =.. s = s for, s =..5 Hit: ObservethatF i, 0 i, is also the umber of rabbits that are at least i moths old at the begiig of the th moth.. s = s s for, s =,s =..7 s = s s s k for k, s i = i for i k,s k = k..8 s =s s s for, s =,s =,s =7..9 Hit: Prove by iductio o..0 Hit: Suppose a = i, i =,,...,. Cosider the cases a i = ada i. k 5 7. s, k i Hit: Collectallthex terms i the expasio of [x] m. ii Hit: Collect all the x m terms i the expasio of [x] m. 7. Hit: Cosider the cases,,,,, ad,,. 7.5 Hit: Remember to show that every arragemet of itegers aroud k idistiguishable circles correspods to a uique permutatio of,,...,. k 5 8. S0, k S5, 8. i Hit: Suppose first that the two groups are distict. ii Hit: First choose items to put i oe group. 8. i Hit: Use iductio o. ii Hit: Use iductio o. 8.5 Hit: First divide the 5 elemets i the domai ito oempty idistiguishable groups.

8 0 Coutig by o 0//8. For persoal use oly. 9. Hit: We establish a correspodece with Problem B as follows. Draw a -go i such a way that the base is horizotal. Label the sides, startig from the left of the base ad goig clockwise, x to x. Leave the base ulabelled at this poit as it will be the last to be labelled. We will ow recursively label the chords ad the base as follows. Fid a triagle where two sides are labelled ad the third side ulabelled. Suppose the two sides of the triagle have bee labelled A ad B, where each subscript i A is less tha ay subscript i B. The the third side will be labelled AB. 9. Hit: We establish a correspodece with Problem C as follows. Label the poits,,..., clockwise i order. For each chord ij, wherei<j,put 0 as the ith digit ad as the jth digit of the correspodig -digit biary sequece. 9. Hit: Divide ito cases accordig to the triagle formed with oe side as the base of the polygo. I ay particular case, the triagle will split the polygo ito two smaller polygos, oe with r sides ad the other with r sides. 9. Hit: Fid the matchig for the first. Now cosider the paretheses withi this pair of paretheses ad the paretheses to the right of it. 9.5 i Hit: Assume the cotrary ad work backwards. ii P = C. Hit: Divide ito cases accordig to the positio of i Stack Z. The use the recursio relatio to show that the sequece is idetical to the Catala umbers. 9. Hit: Expad the biomial coefficiets i ! i ii 0 iii !! ! ii P 00 iii 0 8

9 Aswers to Exercises 05 by o 0//8. For persoal use oly i 5 ii 5 iii 5 5 iv v 5 vi 0 0. i 0 ii 0. i 5, 5, 5, 5, 5, 5 ii! 0. i 7 8! 0.5!!! 0. 5, i 5 ii 7 7 ii i 500 ii iii 0.0 i ii By FTA, express as = p m p m the umber of ways is k i 0 ii 5 0. i 0 ii ! 0.7 i 5 ii 5 iii 5 iv i ii iii 7 7 p m k.the 0.9 i 5 ii 5 iii 5 iv 5 v 5 vi i ii 7 7 iii 8 7 iv v vi k

10 0 Coutig by o 0//8. For persoal use oly. 0. i 0 ii 5 8 iii 0 9 iv v 5 9 vi 9 8 vii 5 8 viii 8 7 ix 0 8 x i 7 ii 7 iii 7 iv 7 5 v 7 0. i ii! iii iv 0. Hit: Cosider the bijectio betwee the set of r-combiatios of N ad the set of -digit biary sequeces with r s, where i the ith positio idicates that i is i the correspodig r-combiatio. 0.5 Hit: Let T i r, for i =,,...,, be the umber of r- combiatios of N i which o two itegers are adjacet aroud the table ad which cotai i i Hit: For =, we have possible graphs. ii i 50 ii 75 iii 95 iv 9 v i 0 ii 0 iii 0 iv 8 v 0 0. i 00 ii 580 iii 0 iv v 0. S =S S for, S =,S = 5. Hit: By iductio. 0. i 5 ii 08 iii 8 0. i Hit: Show two differet ways of distributig r idetical objects ito k idetical boxes so that o box is empty; k =, or = k, or = k. iii,,,, 9 iv Hit: Obtai a bijectio betwee S, the set of all distributios of m idetical objects ito m idetical boxes so that o box is empty, ad T k,fork =,,...,m,thesetof all distributios of idetical objects ito k idetical boxes so that o box is empty. v 7

11 Aswers to Exercises 07 by o 0//8. For persoal use oly. vi Hit: Split the coutig accordig to whether there is a box with exactly oe object. 0.5 i 8; a = a a a a,for 5, a =, a =,a =,a =8. ii b =b b b,for, b =,b =7, b =. 0. i 5 5 ii r r 0.7 i 7 ii iii 9 iv Hit: Objects remaiders i the log divisio of a by b; Boxes remaiders whe divided by b. 0.9 i Hit: Cosider two methods of dividig a group of persos ito three groups cotaiig m, m r ad r persos respectively. ii Hit: Cosider the biomial expasio of x for a particular value of x Further hit: Cosider the cases t<m, t = m ad t>m Hit: LetP be the property divisible by, P be the property divisible by 5, ad P be the property divisible by Hit: Letm = Hit: Geeralise the argumet at the ed of Chapter i Hit: Use mathematical iductio. Alteratively, use the idea ad result of Problem 7.. ii Hit: Ufold the recurrece s m,m=s m, m ms m, m i Hit: Divide ito cases accordig to the umber of persos who are ot i the same group with a particular perso A. ii Hit: Ufold the recurrece relatio S m,m= S m, m ms m, m. 0.5 i {{, }, {, }, {5}}, {{, }, {, 5}, {}}, {{, }, {, 5}, {}}, {{, }, {, 5}, {}}, {{, 5}, {, }, {}}, {{, }, {, 5}, {}}, {{,, 5}, {}, {}}. ii {{, }, {, 5}, {}, {}}, {{, }, {, }, {}, {5}}, {{, }, {, }, {}, {5}}, {{, 5}, {, }, {}, {}}, {{, 5}, {, }, {}, {}}, {{, }, {, 5}, {}, {}}, {{, 5}, {, }, {}, {}}.

12 08 Coutig by o 0//8. For persoal use oly. iii Hit: Divide ito cases accordig to whether the subset {} exists or ot. iv Hit: Use iductio o k i Hit: Cosider the differet umber of oempty subsets that a -elemet set ca be partitioed ito. ii Hit: Divide ito cases accordig to the umber of elemets i the subset of which isaelemet Hit: Obtai the sequece 0 d, d,...,i d i,..., d. Start with the -digit biary sequece Now move the ith 0 to a positio right of i d i s. Show that the resultig sequece fulfils the coditios of Problem C Hit: Obtai a correspodece with Problem C as follows. Give a -digit biary sequece, the itegers i, i =,,...,, are placed i order accordig to the followig rules: O the leftmost empty cell i row A if the ith digit is 0. O the leftmost empty cell i row B if the ith digit is

13 Idex by o 0//8. For persoal use oly. biary sequece,, 8, 0,, 50 biomial coefficiet, 87 Biomial Theorem, 78, 87 Catala umbers, 7 colour, 5 combiatio, deragemet, 5, disjoit, 97 distributio problem, 50, 5, 9 Euler, 7, 7 Fiboacci umbers, 7 Fudametal Theorem of Arithmetic, graph, iteger solutios, 55, 8 Lucas umbers, 5 mappig, 5, 7, 05 bijectio, 7, 50, 55, 7, 79 oe to oe ijectio,, 7, 7, 05 oto surjectio,, 7, 7, 0, 9, Pascal s Triagle, 87 permutatio,,, 7, 5, 57 power set, Priciple Additio,,, 97, Bijectio, 5, 7, Geeralised Pigeohole, Ijectio, 7 Multiplicatio,, 9, 0, of Complemetatio, 0 of Iclusio ad Exclusio, 97 Pigeohole, Ramsey umber, r-combiatio, recurrece relatio, r-permutatio, 0 shortest routes, 9, 0, 7 Tower of Haoi, 9, 09

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