b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

Transcription

1 MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth 23?. How mny trnry wors o lngth 23 with ight 0 s, nin 1 s n six 2 s?. Lt t n not th numr o trnry strings tht o not hv 1 ollow immitly y 2. Fin (ut o not solv) linr rurrn qution stisi y t n. 2. How mny ltti pths rom (2, 3) to (17, 12) pss through (4, 6) n (8, 10)? 3. How mny intgr vlu solutions to th ollowing qutions n inqulitis:. x 1 + x 2 + x 3 + x 4 = 40, ll x i > 0.. x 1 + x 2 + x 3 + x 4 = 40, ll x i 0.. x 1 + x 2 + x 3 + x 4 40, ll x i Us th Eulin lgorithm to in = g(168, 1320).

2 5. Us your work in th pring prolm to in intgrs x n y so tht = 168x y Fin th st o miniml lmnts o this post.. How mny lmnts o r inomprl with th point ll 12?. Explin why {3, 16, 17} is not mximl ntihin.. For h x, lt hight(x) not th mximum siz o hin hving x s its grtst lmnt. Writing irtly on th igrm, ll h point with th intgr rprsnting its hight.. Fin th hight h o this post. Fin hin o h points.

3 7. g. This post is n intrvl orr n hs 5 istint own sts. Fin thm.. This post lso hs 5 istint up sts. Fin thm.. Fin th uniqu intrvl rprsnttion or this post whr vry lmnt is ssign n intrvl with intgr npoints rom {1, 2, 3, 4, 5}. 8. Din n intrvl orr P with point st X = {,,,,,, g, h, i, j}. y th ollowing intrvl rprsnttion. j h i g Us th First Fit lgorithm to prtition o this post into minimum numr o hins. Provi your nswr y lling th intrvls in th igrm with positiv intgrs so tht ll lmnts ssign th sm intgr orm hin. Thn in mximum ntihin in this post.

4 9. k j g h l i Us th Gry Algorithm n lphti orr to in n ulr iruit in th grph ov. Your nswr shoul givn s squn o prtil iruits strting with th trivil iruit () h g In th sp low, list in orr th gs whih mk up minimum wight spnning tr using, rsptivly Kruskl s Algorithm (voi yls) n Prim s Algorithm (uil tr). For Prim, us vrtx s th root. Kruskl s Algorithm Prim s Algorithm

5 Show tht this grph is hmiltonin y listing th vrtis in n orr whih orms yl o siz 10.. Explin why this grph hs nithr n ulr iruit nor n ulr pth. 12. A t il igrph t.txt hs n r or igrph whos vrtx st is [6]. Th wights on th irt gs r shown in th mtrix low. Apply Dijkstr s lgorithm to in th istn rom vrtx 1 to ll othr vrtis in th grph. Also, or h x, in shortst pth rom 1 to x. W

6 13. Writ th gnrl solution o th vnmnt oprtor qution: (A 2) 3 (A 1) 4 (A + 6) 2 (A 8) = Fin prtiulr solution to th vnmnt oprtor qution: (A 2 9A + 18)(n) = 20(2) n. 15. Fin th uniqu solution to th vnmnt oprtor qution: (A 2 9A + 18)(n) = 20(2) n with (0) = 3 n (1) = Lt X st n lt P = {P 1, P 2,..., P m } mily o proprtis. For h sust S {1, 2,..., m}, lt N(S) not th numr o lmnts o X whih stisy proprty P i whnvr i S. Writ th Inlusion-Exlusion ormul or th numr o lmnts o X whih stisy non o th proprtis in P: 17. Writ th Inlusion-Exlusion ormul or th Eulr-φ untion. 18. Us th ormul rom th pring prolm to in φ(n) whn n = Lt R(n, m) not th lst positiv intgr t so tht vry grph on t vrtis ontins omplt sugrph o siz n or n inpnnt st o siz m. Bo lims tht R(3, 3) = R(4, 4) = 6. Ali rplis tht Bo is only hl right. R(3, 3) = 6 ut R(4, 4) > 6. Explin why Ali s ssrtion tht R(4, 4) > 6 is orrt.

7 20. Wht is th ormul or th numr o ll trs with vrtx st {1, 2,..., n}? 21. How mny wys r thr to ssign lls rom th st {1, 2,..., 10} to th unll tr shown low? 22. S 84, 60 63, 25 17, 17 G C E 38, 5 43, 43 39, 9 43, 17 6, 6 28, I A H 78, 43 30, 30 7, 7 24, 24 B 30, 5 18, J D F 59, 31 5, 5 90, 66 T. Wht is th urrnt vlu o th low?. Wht is th pity o th ut V = {S, A, C, I, E, G, H} {B, D, F, J, T }.. Crry out th lling lgorithm, using th psuo-lphti orr on th vrtis n list low th lls whih will givn to th vrtis.. Us your work in prt to in n ugmnting pth n mk th pproprit hngs irtly on th igrm.

8 . Crry out th lling lgorithm son tim on th upt low. It shoul hlt without th sink ing ll. Fin ut whos pity is qul to th vlu o th low. 23. In th igur ov, w show post n th iprtit grph ssoit with it. Th rkn gs orm mximum mthing in th grph. Fin th minimum hin prtition trmin y this mthing.