MS2005/MS6005 Exercise Sheet

Transcription

1 10/9/15 MS2005/MS6005 Exris Sht 1. In ah as omput th numbr of intgrs that ar multipls of 11 in th givn st: A = {10,11,12,...,54}, B = {11,12,13,...,55}, C = { 127, 126,...,39} 2. How many intgrs in th st {4,5,6,...,91,92} ar multipls of 5? How many ar multipls of 7? How many ar multipls of nithr 5 nor 7? 3. Th numbr N = 1000! = ns in xatly how many zros? 4. Prov th inlusion-xlusion prinipl for thr finit sts A, B an C: A B C = A + B + C A B B C C A + A B C Formulat th inlusion-xlusion prinipl for 4 finit sts. 5. Consir a group of 191 stunts of whih 10 blong to Chss Club, Law Soity an Suas; 36 blong to Chss Club an Law Soity; 20 blong to Chss Club an Suas; 18 blong to Law Soity an Suas; 65 blong to Chss Club; 76 blong to Law Soity; an 63 blong to Suas. (i) How many blong to Chss Club an Suas but not Law Soity? (ii) How many blong to Law Soity but nithr Chss Club nor Suas? (iii) How many blong to Suas or Chss Club (or both) but not Law Soity? (iv) How many blong to non of th soitis? 6. All stunts in an arts ollg partiipat in at last on out of thr ativitis: painting, musi, an film lub. If 10 stunts o all thr ativitis, 150 stunts o two or mor ativitis, half of all stunts partiipat in musi, an th ratio of partiipants in painting, musi, an film lub, rsptivly, is 5:4:3, what is th total numbr of stunts in arts? 7. How many tims os th igit 7 our in th list of numbrs from 1 to 999? How many of numbrs from 1 to 999 hav at last a igit of 7? 8. How many 8-bit binary strings (i) bgin an n with 1? (ii) hav ithr th son or fourth bit 1 (but not both)? (iii) hav xatly on 1? (iv) hav xatly two 1 s? (v) hav at last on 1? (vi) ra th sam from ithr n (.g. 1010, but not 1100)? 9. Eah usr on a omputr systm has a passwor whih is four to six haratrs long, whr ah haratr is an uppras lttr or a igit How many passwors ar thr if ah passwor must ontain both lttrs an igits?

2 10. Th lttrs ABCDEF ar to b us to form strings of lngth 3. (i) How many strings an b form if w allow rptitions? (ii) How many strings an b form if w o not allow rptitions? (iii) How many strings bgin with A, allowing rptitions? (iv) How many strings bgin with A if rptitions ar not allow? (v) How many strings o not ontain A if rptitions ar not allow? (vi) How many strings ontain th lttr A, allowing rptitions? (vii) How many strings ontain th lttr A if rptitions ar not allow? 11. A six prson ommitt ompos of Aoif, Barry, Ciara, Dlan, Eoin an Fiona is to slt a hairprson, srtary an trasurr. (i) How many sltions ar thr in whih nithr Barry nor Fiona is an offir? (ii) How many sltions ar thr in whih both Barry an Fiona ar offirs? (iii) How many sltions ar thr in whih ithr Dlan is hairprson or h is not an offir? (iv) How many sltions ar thr in whih Barry is ithr hairprson or trasurr? 12. How many strings an b form by orring th lttrs ABCDE: (i) ontaining th substring ACE? (ii) ontaining th lttrs A, C an E togthr in any orr? (iii) ontaining ithr th substring AE or th substring EA? (iv) ontaining nithr of th substrings AB, CD? (v) suh that A appars bfor C an C appars bfor E? 13. How many trms ar thr in th xpansion of (x+y)(a+b+)(+f +g)(h+i)? 14. In how many ways an 6 popl b arrang aroun a irular tabl in ah of th following ass? (i) An arrangmnt onsists in hoosing ah prsons s nighbors. Th atual sats on t mattr. (ii) Thr ar 3 womn an 3 mn an thy n to b arrang so that womn an mn altrnat. Again, ompany, not th atual sating, mattrs. 15. A library has just riv 4 iffrnt art history books, 6 iffrnt Irish languag books an 5 iffrnt molular biology books. (i) In how many ways an ths books b arrang on a shlf? (ii) In how many ways an ths books b arrang on a shlf if all th molular biology books ar to b group togthr? (iii) In how many ways an ths books b arrang on a shlf if all books of th sam isiplin ar group togthr?

3 16. Th Irish Lotto rquirs a partiipant to hoos six numbrs from among 45 (thir orr osn t mattr). In how many ways an this b on? 17. A rtain lub onsists of six mn an svn womn. (i) In how many ways an w slt a ommitt of fiv popl? (ii) In how many ways an w slt a ommitt of four popl that inlus at last on woman? (iii) In how many ways an w slt a ommitt of four popl that has rprsntation of both sxs? 18. A brig han onsists of 13 ars takn from a stanar pak of 52 ars. (i) How many brig hans ar thr? (ii) How many brig hans ar thr with all ars of th sam suit? (iii) How many brig hans ontain xatly two suits? (iv) How many brig hans ontain all four as? (v) How many brig hans ontain fiv ars of on suit, four of anothr suit, thr of anothr suit an on of anothr suit? 19. How many routs ar thr from th lowr-lft ornr of an m n gri to th upprright ornr if w ar rstrit to travlling only right or upwar to th nxt vrtx? How many of ths routs avoi th point (2,2)? 20. How many strings an b form in ah as by orring th givn lttrs: (i) GUIDE (ii) SCHOOL (iii) SALESPERSONS (iv) SALESPERSONS if no two S s ar onsutiv (v) SALESPERSONS if all four S s ar onsutiv (vi) SALESPERSONS if xatly thr S s ar onsutiv (vii) SALESPERSONS if thy ontain two pairs of two onsutiv S s (viii) SALESPERSONS if thy ontain xatly on pair of two onsutiv S s 21. DNA squns ar strings ma up of th lttrs A, C, G an T. How many strings of lngth 30 ar thr with: (i) 10 A s, 5 C s, 12 G s an 3 T s? (ii) qual numbrs of A s an T s, an qual numbrs of C s an G s? (iii) twi as many A s as C s, an thr tims as many G s as T s? (iv) 10 A s, 5 C s, 12 G s an 3 T s, if no two A s ar to appar onsutivly? (Hint: first mak a squn of C s, G s an T s, thn pla th A s in spas in btwn). 22. Consir pils ofr, bluangrnballs(ahpilontains atlast 10balls):

4 (i) How many ways an 10 balls b slt? (ii) How many ways an 10 balls b slt if at last on r ball must b slt? (iii) How many ways an 10 balls b slt if at last on r ball, at last two blu balls an at last thr grn balls must b slt? (iv) How many ways an 10 balls b slt if xatly on r ball an at last on blu ball must b slt? (v) How many ways an 10 balls b slt if twi as many r balls as grn balls must b slt? 23. Fin th numbr of intgr solutions to th quation x 1 +x 2 +x 3 = 15 subjt to th givn onitions: (i) x 1 0, x 2 0, x 3 0 (ii) x 1 1, x 2 1, x 3 1 (iii) x 1 0, x 2 > 0, x 3 = 2 (iv) 0 x 1 6, x 2 0, x In how many ways an 15 intial appls b istribut among 6 stunts? 25. Show that for any intgrs n 1,M 0, th numbr of nonngativ intgr solutions to th inquality x 1 + +x n M is C(M +n,n). 26. Thr fair, stanar six-fa i of iffrnt olors ar roll. In how many ways an th i b roll suh that th sum of th numbrs roll is: (i) 10? (ii) 15? 27. Fin th offiint of th givn trm whn th xprssion is xpan: (i) x 4 y 7 ; (x+y) 11 (ii) s 6 t 6 ; (2s t) 12 (iii) x 2 y 3 z 5 ; (x+y +z) 10 (iv) w 2 x 3 y 2 z 5 ; (2w+x+3y +z) 12 (v) a 2 x 3 ; (a+x+) 2 (a+x+) 3 (vi) a 2 x 3 ; (a+ax+x)(a+x) Evaluat th following sums: (i) (iv) (vii) (x) 3 (2j +1) (ii) 4 C(4,j) 5 C(5, i) i=3 3 ( 1) j C(5,j) (v) (viii) (xi) (2j +1) (iii) 2 j C(5,j) 2 C(5,i) i=0 2 C(5, 2j) (vi) (ix) (xii) 7 C(7, j) 7 C(7, i) i=4 2 C(5,2j +1) Do you noti any pattrns? Can you xplain thm? 29. Us th intity C(n+1,k) = C(n,k 1) + C(n,k), vali for 1 k n 1, an inution to prov th Binomial Thorm.

5 30. (a) Lt m,n 1 b intgrs an r an intgr suh that r m an r n. By onsiring ways of onstruting r-lmnt substs of an (m + n)-lmnt st, show that r ( )( ) ( ) m n m+n = k r k r k=0 (b) Show that for all intgrs n 1, k=0 ( ) 2 n = k ( ) 2n. n 31. Fin th sums n, (n 1)n, an n 2. [Hint: what ar C(n,1) an C(n,2)? Apply ths to an intity prov in th lturs.] 32. Prov that n2 n 1 = kc(n, k). [Hint: us th Binomial Thorm an alulus. k=1 Altrnativly, ount in how many way on an hoos a ommitt with a hairprson from among n popl.] 33. Giv two proofs that for any 1 k n w hav kc(n,k) = nc(n 1,k 1), on proof bing algbrai, th othr ombinatorial (hint: onsir hoosing a k lmnt subst with a istinguish lmnt from an n lmnt st). 34. Us inution to prov th following: (i) i 2 = 1 n(n+1)(2n+1) for vry n 1 6 (ii) i=1 i 3 = 1 4 n2 (n+1) 2 for vry n 1 i=1 (iii) n 3 +2n is a multipl of 3 for vry n 1 (iv) 3 n < n! for vry n > 6 (v) 11 n 6 is ivisibl by 5 for vry n Dtrmin whih amounts of postag an b pai using just 5 an 6 nt stamps. Prov your answr using inution. 36. Lt a b th squn fin by a n = n 2 3n+3 for n 1. Calulat th following: (i) 4 i=1 a i (ii) 5 ja j j=3 (iii) 4 i=4 a i (iv) 6 k=1 a 2 k+1 (v) 3 i=1 a i (vi) 4 x=3 a x 37. (a) Lt a = (a n ) n=0 b a squn. Simplify N n=1 (a n a n 1 ). (b) By writing 1/k(k +1) = 1/k 1/(k +1), valuat 100 k=11/k(k +1). () Using 2k 1 = k 2 (k 1) 2, valuat n k=1 (2k 1), an hn valuat n k=1 k by an altrnativ mtho to that us in th ltur. 38. Fin th first fiv trms of th squn fin by ah of ths rurrn rlations

6 an initial onitions: (i) a n = 2a n 1, a 0 = 1 (ii) b n = b n 1 b n 2, b 0 = 2,b 1 = 1 (iii) t n = nt n 1 +t 2 n 2, t 0 = 1,t 1 = 0 (iv) C n = C k 1 C n k, C 0 = For ah of th following squns fin a rurrn rlation an initial onitions that gnrat a squn that bgins with th givn trms: (i) 3,7,11,15,... (ii) 3,6,9,15,24,39,... (iii) 1,1,2,4,16,128,4096, Th Fibonai squn is fin by th rurrn rlation: k=1 f 0 = 1, f 1 = 1, f n = f n 1 +f n 2, n 2. So f 2 = 2, f 3 = 3, f 4 = 5, t. (a) Us inution to prov th following: (i) f k = f n+2 1, n 0 k=0 (ii) f 2 n = f n 1f n+1 +( 1) n, n 1 (iii) fk 2 = f n f n+1, n 0 k=0 (iv) f n > ( 3 2) n, n 5 (b) Fin an xpliit xprssion for f n by solving th rurrn rlation. 41. Th squn (g i ) i=0 is fin by th rurrn rlation g n = g n 1 +g n 2 +1, n 2, an g 0 = 1, g 1 = 1. Show using inution, or othrwis, that g n = 2f n 1 for ah n 0, whr f n is th Fibonai squn from th prvious qustion. 42. Solv th following rurrn rlations, with th givn initial onitions: (i) a n = 3a n 1, with a 0 = 2. (ii) a n = 2na n 1, with a 0 = 1. (iii) 6a n 1 9a n 2, with a 0 = 1, a 1 = 3. (iv) a n = 2 n a n 1, a 0 = 1. (v) a n = 6a n 1 8a n 2, a 0 = 1, a 1 = 0. (vi) a n = 7a n 1 10a n 2, a 0 = 5, a 1 = 16. (vii) a n = 2a n 1 +8a n 2, a 0 = 4, a 1 = 10. (viii) 2a n = 7a n 1 3a n 2, a 0 = a 1 = 1. (ix) a n = 2a n 1 2a n 2, a 0 = 2, a 1 = 6. (x) an = a n 1 +2 a n 2, a 0 = a 1 = A ar alr wants to isplay motoryls an ars in a row of n spas. Eah yl rquirs on spa an ah ar ns two. (All th motoryls look th sam an th sam is tru for all th ars). Fin an solv a rurrn rlation for th numbr of iffrnt isplays possibl.

7 44. Fin th numbr of ways in whih a 2 n strip an b fill with tils of sizs 1 2 an Fin th numbr of ways in whih a 2 n strip an b fill with tils of sizs 1 1 an 1 2, suh that no pairs of 1 2 tils in th following positions ar to b foun in th pattrn: 46. Thirtn popl hav first nams Darrn, Emma an Maira, an surnams Dwyr, Man, O Connor an Shhan. Show that at last two popl hav th sam nam. 47. An invntory onsists of 115 itms, ah mark availabl or unavailabl. Thr ar 60 availabl itms. Show that thr ar at last two itms in th list xatly four apart. 48. Suppos a party has six popl. Consir any two of thm: thy might b mting for th first tim, whih as w will all thm mutual strangrs; or thy might hav mt bfor, whih as w will all thm mutual aquaintans. Show that ithr at last thr of thm ar (pairwis) mutual strangrs or at last thr of thm ar (pairwis) mutual aquaintans. 49. A 3 7 rtangular boar is ivi into 21 squars ah of whih is olour r or grn. Prov that th boar ontains a rtangl (of minimum hight an with 2) whos four ornr squars ar all olour r or grn. 50. Showthatvryintgrnhasamultiplonsistingonlyof0 san1 s. (Hint: onsir what happns whn you ivi th n+1 numbrs 1, 11, 111,..., } 11 1 {{} by n.) n A oin is toss an a i is roll. (i) List th mmbrs of th sampl spa. (ii) List th mmbrs of th vnt Th oin oms up has, an th i is an vn numbr. (iii) Assuming th oin an i ar fair, what is th probability that th oin oms up tails, an th i a numbr lss than 4? 52. Whn typing a rport, a typist maks i rrors with probability p i, (i 0), whr p 0 = 0.20, p 1 = 0.35, p 2 = 0.25 an p 3 = What is th probability that th typist maks (i) at most two rrors; (ii) at last four rrors? 53. Two fair i ar thrown. Fin th probability that (i) at last on of th i oms up as 5; (ii) at last on of th i oms up vn;

8 (iii) at last on of th i oms up ithr 3 or 5; (iv) at last on of th i oms up ithr 3 or vn. Romput ah part with at last hang to xatly. 54. Whih of th following is mor probabl: (i) gtting at last on 6 with four throws of a i, or (ii) gtting at last on oubl 6 with twnty-four throws of two i? Assum that th i ar fair. (This is somtims all Méré s paraox, aftr th profssional gamblr Chvalir Méré who bliv ths two vnts to hav qual probability.) 55. Four miroprossors ar ranomly slt from a bath of 100, among whih 10 ar ftiv. Fin: (i) Th probability of obtaining no ftiv miroprossors. (ii) Th probability of obtaining xatly on ftiv miroprossor. (iii) Th probability of obtaining at last on ftiv miroprossor. 56. An unprpar stunt taks a tst onsisting of 10 tru-or-fals qustions, an so gusss th answr to ah qustion, with an qual liklihoo of gtting th answr right or wrong. (i) What is th probability that th stunt answrs vry qustion orrtly? (ii) What is th probability that th stunt answrs vry qustion inorrtly? (iii) What is th probability that th stunt answrs xatly on qustion orrtly? (iv) What is th probability that th stunt answrs xatly fiv qustions orrtly? 57. A han of 5 ars is alt from a pak of 52. Fin th probability that th han ontains i kings, for i = 0,1,2,3 an 4. What is th sum of ths fiv probabilitis? 58. Loa i ar rat so that 2, 4 an 6 ar qually likly to appar, an 1, 3 an 5 ar qually likly to appar, but 1 is thr tims mor likly to appar than 2. (i) On i is roll. What is th probability of gtting a 5? (ii) On i is roll. What is th probability of gtting an vn numbr? (iii) On i is roll. What is th probability of not gtting a 5? (iv) Two i ar roll. What is th probability of gtting oubls? (v) Two i ar roll. What is th probability of gtting a sum of 7? (vi) Two i ar roll. What is th probability of gtting a sum of 6 givn that at last on i shows 2?

9 59. Show, using inution, that for any squn of vnts A 1,A 2,A 3,... w hav P(A 1 A 2 A n ) P(A i ). i=1 60. Suppos thata, B anc arvnts inasampl spas. UsthformulaP(E F) = P(E)+P(F) P(E F), with E = A B an F = C, to show that P(A B C) = P(A)+P(B)+P(C) P(A B) P(B C) P(A C)+P(A B C) 61. Thr i ar thrown. Fin th probability that at last on i oms up 3. [Hint: on mtho involvs using th rsult from th prvious qustion. Othr rasonabl mthos also xist.] Fin th probability that at last two i om up Show that P(A 1 A 2 A 3 ) = P(A 3 A 1 A 2 )P(A 2 A 1 )P(A 1 ) for any possibl hoi of vnts A 1, A 2 an A 3 that satisfy P(A 1 A 2 ) > An urn ontains r r balls an b blu balls (at last 3 of ah). Balls ar rawn from th urn without rplamnt. Fin th probability that (i) th first ball rawn is r; (ii) both th first an son balls rawn ar r; (iii) th first, son an thir balls rawn ar all r; (iv) th first ball is r an th son is blu; (v) th first ball is blu an th son is r; (vi) th son ball is blu. 64. In a fatory prouing ompat iss, th total quantity of ftiv itms foun in a givn wk is 14%. It is suspt that th majority of ths om from two mahins, X an Y. An insption shows that 8% of th output from X an 4% of th output from Y is ftiv. Furthrmor, 11% of th ovrall output am from X an 23% from Y. A CD is hosn at ranom an foun to b ftiv. What is th probability it was ma by mahin X or Y? 65. A fair oin is toss 10 tims. Fin th probability that has oms up xatly fiv tims. Fin th probability that has oms up at most fiv tims. 66. A oin is toss 10 tims. Fin th probability that has oms up at most four tims, if th probability that w gt has on any givn toss is 1/ JoanSinatakatst. ThprobabilitythatJopasssis0.75,anthprobability that Sina passs is 0.8. Assum that th vnts Jo passs an Sina passs ar inpnnt. Calulat th following: (i) Th probability that Jo os not pass. (ii) Th probability that both pass. (iii) Th probability that both fail. (iv) Th probability that at last on of Jo or Sina passs.

10 68. A oin is toss rpatly. At ah toss th probability of has oming up is p. (i) Fin th probability that th first ha oms on th first toss; (ii) Fin th probability that th first ha oms on th son toss; (iii) Fin th probability that th first ha oms on th thir toss; (iv) Fin th probability that th first ha oms on th ith toss. Philosophial qustion: is it possibl that you nvr gt a ha? 69. Suppos that A an B ar inpnnt vnts. Show that A an B ar inpnnt; that A an B ar inpnnt; an that A an B ar inpnnt. 70. (a) Suppos S is a sampl spa an that thr ar vnts A 1,A 2,...,A n that partition S. That is, ah outom from S blongs to xatly on of th vnts A i. Lt B b any vnt. Prov Bays Thorm: P(A j B) = P(B A j)p(a j ) n i=1 P(B A i)p(a i ) for ah 1 j n (b) Dlan, San an Louis work in a tlmarkting ntr. Th following tabl shows th prntag of alls ah allr maks an th prntag of prsons who ar annoy an hang up on ah allr: Dlan San Louis % of alls % of hang-ups Lt D not th vnt Dlan ma th all, S th vnt San ma th all, L th vnt Louis ma th all an H th vnt th allr hung up. Fin P(D), P(S), P(L), P(H D), P(H S), P(H L), P(D H), P(S H), P(L H) an P(H). 71. In a variant of th Monty Hall problm, a ontstant is ask to hoos on of four oors; bhin thr of th oors thr ar goats, an bhin th fourth oor is a ar. Aftr th ontstant hooss a oor, th host piks on of th othr thr oors, on that his a goat, an opns it. Th host thn givs th ontstant th option of abanoning th hosn oor in favour of on of th two still-los, unhosn oors. For ah of th following stratgis, what is th probability of winning th ar? (i) Stay with th initial hoi. (ii) Choos ranomly btwn th oor first hosn an th two still-los oors. (iii) Swith to a ranom hoi of on of th two still-los oors? 72. Th intrstion graph of a olltion of sts A 1, A 2,..., A n is th graph that has a vrtx for ah of ths sts an has an g onnting th vrtis rprsnting two sts if ths sts hav a nonmpty intrstion. Construt th intrstion graph of th following olltions of sts: (i) A 1 = {0,2,4,6,8}, A 2 = {0,1,2,3,4}, A 3 = {1,3,5,7,9}, A 4 = {5,6,7,8,9}, A 5 = {0,1,8,9} (ii) A 1 = {..., 4, 3, 2, 1,0}, A 2 = {..., 2, 1,0,1,2,...} = Z, A 3 = {..., 6, 4, 2,0,2,4,6,...}, A 4 = {..., 5, 3, 1,1,3,5,...}, A 5 = {..., 6, 3,0,3,6,...}

11 (iii) A 1 = {x : x < 0}, A 2 = {x : 1 < x < 0}, A 3 = {x : 0 < x < 1}, A 4 = {x : 1 < x < 1}, A 5 = {x : x > 1}, A 6 = R 73. Draw th aquaintanship graph that rprsnts that Ptr an Lina, Ptr an Mihll, Ptr an Sarah, Ptr an Ciara, Ptr an Aoif, Brian an Lina, Brian an Eoin, Lina an Mihll, Ciara an Mihll, an Ciara an Aoif know ah othr, but that non of th othr pairs of popl list know ah othr. (That is, two popl ar ajant if thy ar aquaintans.) 74. Fin th numbr of vrtis, th numbr of gs, an th gr of ah of vrtx in th givn graphs. Fin th sum of th grs of th vrtis an vrify that it quals twi th numbr of gs. a a b (i) (ii) b f f 75. Can a simpl graph xist with 15 vrtis ah of gr 5? 76. Show that th sum, ovr th st of popl at a party, of th numbr of popl a prson has shakn hans with, is vn. Assum that no-on shaks hans with thmslvs. 77. Draw th following graphs: (i) K 7 (ii) K 1,8 (iii) K 4,4 (iv) Q Dtrmin whih of th following graphs ar bipartit. a b a a b (i) (ii) b (iii) f 79. Dtrmin thos valus of n an m for whih th graphs K n, K m,n an Q n ar bipartit. 80. How many gs an how many vrtis ar thr in th graphs K n, K m,n an Q n? 81. How many gs os a graph hav if its vrtis hav gr 4,3,3,2,2? Draw suh a graph. 82. Dos thr xist a graph with six vrtis of th givn grs? If so, raw suh a graph: (i) 0,1,2,3,4,5 (ii) 1,2,3,4,5,6 (iii) 2,2,2,2,2,2 (iv) 3,2,3,2,3,2 (v) 3,2,2,2,2,3 (vi) 1,1,1,1,1,1 (vii) 3,3,3,3,3,5 (viii) 1,2,3,4,5,5 83. Lt G b a graph with v vrtis an gs. Lt M b th maximum gr of

12 th vrtis of G, an lt m b th minimum gr of th vrtis of G. Show that m 2/v M. 84. Fin th numbr of paths of lngth n btwn any two ajant vrtis in K 3,3 for n = 2,3,4 an 5. Fin th numbr of paths of ths lngths btwn two nonajant vrtis. 85. Dtrmin whih of th following graphs ontain Eulr iruits, an onstrut thm whn thy xist. If no Eulr iruit xists, trmin whthr th graph has an Eulr path, an onstrut it if it xists. a a b (i) b (ii) f a g h i (iii) 0 1 b f 86. For whih valus of m an n os th omplt bipartit graph K m,n hav an Eulr iruit? An Eulr path? 87. Dtrmin whih of th following graphs hav a Hamilton iruit, an fin it if it xists. If it os not xist, xplain why not. a a b (i) (ii) b f a a b (iii) (iv) b f

13 88. Fin th lngth of th shortst path btwn a an h, btwn a an, btwn a an f, btwn an f, an btwn b an h. In ah as giv a path that is of this minimum lngth. 4 b f a h g Explain how to fin a path with th last numbr of gs btwn two vrtis in a graph by onsiring it as th shortst path problm in a wight graph. 90. Solv th Travlling Salsman Problm for th following graph by fining th total wight of all Hamilton iruits. 7 a b Fin th shortst ommon suprstring whih has th following lngth 3 substrings: S = {AAC,ACC,ACG,CCA,CCG,CGC,CGT,GCC,GTA,TAC}. Solv this problm by fining an Eulr path using th strings in S as labls of gs. Is your solution uniqu? 92. Fin th shortst ommon suprstring whih has th following lngth 3 substrings: S = {AGT,AAA,ACT,AAC,CTT,GTA,TTT,TAA}. Solv this problm using two iffrnt approahs: by fining a Hamilton path using th strings in S as th vrtis, an by fining an Eulr path using th strings in S as labls of gs. (For th son approah you will hav to al with igraphs that hav loops.) 93. Fin th shortst ommon suprstring for all 8 = bit binary strings. Rpat th task for all 16 4-bit binary strings. 94. Fin th shortst ommon suprstring for th following DNA fragmnts: Is th solution uniqu? S = {ATTAC, CAGG, CGTA, TATCA}.