Midterm - UNIVERSITY OF WATERLOO MIDTERM EXAMINATION SPRING...

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Unformatted text preview: UNIVERSITY OF WATERLOO MIDTERM EXAMINATION SPRING TERM 2006 Surname: L First Name: L‘ with L Course Number MATHEMATICS 239 Course Title Introduction to Combinatorics U/P. Schellenberg 9:30 MWF LEC 001 Instructor 1:: U. Celmins 1:30 MWF LEC 002 Date of Exam 7 Tuesday , June 27, 2006 Time Period 4:30 - 6:30 p.m. Number of Exam Pages 10 (including this cover sheet) Exam Type 1 Closed Book Additional Materials Allowed None Additional Instructions 1. Write your answers in the space provided. If you require more space, use the back of the previous page. 2. Please indicate your professor and section above. Mark Awarded Mark Awarded ‘ 4b 5 r; X MATH 239 Mid-Term Exam, Jun. 27, 2006 2 1. [4 marks] Determine [51:10] ’33 W' _ K l *5" ,1 s 51‘ 4/ '> 4'"— %§@X) - ’K {1/ /7Z r M ,4 V” . V 4 /" . :' r L9 1,» Que, mi! - -—w ' 1 {gift ' (26> : [/th ( t L}; _ 2 (b) [4 marks] Determine [$13]%5%3. Express your answer as a sum of products of bino- mial coefficients. //v 2 >8 a — ; 1’ - (/v 29634 ( F 7&7" v 5 K 0“ W ’v/ (J, : E/k/CZZ") €( i)’ /<-‘0 Mia . ' co 9;. J g {fflflkxm / 9”]ij S" (i>(?:]>(‘2)k J: c3~2£< ZKHUS ‘ M: (2:; £ 6 we-“ , k \13 j 8 ( M M K19 ‘ J'_ a, / MATH 239 Alid—Term Exam, Jun. 27, 2006 (c) [5 marks] Use the Binomial Theorem to pr0ve that (“I “ I) (’3) = (“"7”)- : B5”! g ( Thwkw‘ffl" ( ’3’" 25/ "1“??sz (/«Fijw (/«P)C>W , (W ’0») MATH 239 Mid—Term Exam, Jun. 27, 2006 4 2. [4 marks] List all the 3—part compositions of 5. (Recall that compositions have no ‘ zeros.) /' (940W 3~ I (21 l 12:.) y (b) [5 marks] Determine the generating function for the number of compOsitions of n in which every part is an even positive integer. fizz/4 " -(%‘T2 W";_. . ): fl‘z __,..,—-—-'— 7. //Z’ 3‘5"}: /Z(vv- Vv‘l ,“X‘L : /—%Z/(/"%L) Z"? flaw/“Dal #2"; (L V ’ M29. “12/ /”(77(‘1 / MATH 239 Mid—Term Exam, Jun. 27, 2006 5 3. (a) [4 marks] Write down a decomposition that uniquely creates all binary strings in which each block of 1’s has length at least 3. 3H? 0% ‘ztéfll (é U Sem'rlfi 073 WWW 7‘ a k [5 marks] Let A be the set of binary strings defined as follows: A = {01, 0011, 000111, 00001111, . . La 2 9 Determine the generating function for the binary strings in the set 5 = {1}*({0}*A)*{0}*- As usual, the weight of a string is its length. ;;(r>f ‘17? fl‘ t‘ (603%)”? ' xxx( ‘*p;;;1 0rfii Z: <:§I+w / ~ 76 MW“ “M - 3 IIZxJZflt’Y {fl 2 / r >61 ,7: j (L x) 3* 2: 2 / w ,1: 7.. )6 f \, WM_MW__ m, i v I V" 3 Z /“ 1 [*Er-Yt—lzl'z‘ '*)"7% 8 MATH 239 Mid—Term Exam, Jun. 27, 2006 6 4. (a) [4 marks] The generating function for an, the number of binary strings of length n that do not contain the substring 01110, is Za‘$i_ 1+$4 t " _ 4_ 5 20 1 2w+w a; Find a linear recurrence relation satisfied by the an’s, together with sufficient initial conditions to uniquely determine the sequence {an}. (bawdy/76’) $6») = H2” [x0] 40 \ C1,?i 4/9740 1'. O / MATH 239 Mid—Term Exam, Jun. 27, 2006 (b) [5 marks] Let the sequence bn be defined by b0 1: 3, bl = 4, b2 2 12 and Solve this recurrence relation to obtain a. closed form expression for bn. y/[ff [A4V0l/4V/j??? /J/;lw‘Vi/i/¢ / /J" /%U figvé%:tlle’8 ~, WU: /, £49 ".5 X /[&):5*ZY%27,5 ;& '7c~2 \. ,,,_,-.~‘ ) ,. WNWKMM.» . / 751 ’“ ‘f (film-W www.mh. 70-2 7/2/92? flzx «9 W? ,276: 2 X : (76v?)3 « “9%L442x/ \\ “‘(x‘ + 5 ,6 Mr VXLr a biz v o w ,,,,,,, ,mé n f_,,‘ 0”]! #01171 , 2-. “#7; MAI/’7 3 _‘ / V1 1 . g” (4 we“ f 6/41) '4 z WVWW M“— é0:3: {“A‘ZaZA' (Ari/5%); :LMWC w w \ H 4: \\ V W 1 , 2 7 ( I ( n‘,_’,,vwc / _ v W 1 L( I > ’ 1 + Z.C ( r \ ("SCI E V7 {2 7: (7,6 “206% m 0/) MATH 239 Mid-Term Exam, Jun. 27, 2006 . 8 y 5.‘ (a) [4 marks] Make a list of all graphs, up to isomorphism, having 5 vertices and 4 edges; that is, each graph with 5 vertices and 4 edges should be isomorphic to exactly one of the graphs on your list. (Hint: There are more than 4 and fewer than 8 such graphs.) ' 5 / (fl-4V. * M (g > [701 MM f/jaf ‘ Z “A 2 . . (b) [4 marks] Show that if every vertex of a graph G has degree at least 1:, then G has a path of length at least k. (Hint: Let 09,111,02, . . . ,1)" be a longest path in G’. Show that 'n. 2 k.) / [/2 ‘ W [/14 J, a fl/z/ ox/ /,//é’/44 ,1) 6. w V‘ [VWh-‘fl [fl ‘7z k» 97/4fr :— éwamc 1/,4. "/7 [mazsz 72/) 47¢i/qu k‘ WV/lx/i way-7324f; a [OW/lél/M 740 kvkal MVf/‘(eJ not Fm (/1. [la/(y! %éI/” 44/ flvly __ [M < AVI’WI/fl yet/(fr _ an. o 7%,? fin/NH] flay 3/4”! (ME?) fame MOZX “4+; [74 9’ //7é77‘%o/L' ‘4, 752577 a 7 7 Mg flan/M 6/ // 2M ///W/i/iwé77497 m WWW/7W W/ /1 a flaw. a Mix/L7 azkfi . \é w AIATH 239 Mid-Term Exam, Jun. 27, 2006 9 6. For each integer n 2 1, let An be the graph whose vertices are all subsets of {1,2, . . . , 2n} having n or n + 1 elements, and two distinct vertices (subsets) are adjacent if and only if one contains the other. For example, the vertex {1,2, 3,4} is adjacent to the vertex {1, 2,4} in A3. (a) [4 marks] Determine the number of edges in An. ? 4,, LA ~ 2‘ W 4 2'“ 8/42/61 it _ "“ VIA 4 l 0 I r >(L (12f fl/IJIVKKY‘HI 9’”? f?'/ (M79141! oft/flap fl; tgffixé/ef 5’1 [fit 63¢ x xxx/1 (fl%’/)W6//e”1< [am/M”? (Mt/J 67/4” *W/é’czé -%/%W/:' 6; r-‘mxpu/w/ ‘7 9’74" Z/fl‘VMW/ W! f”, a? (an, 740241,! W (“fly ’Mfltk- 7'25? V5: apt {42.6 / {[flr’tz 6': f? R 1/»! ("W I) ~ tiff/x, a?on :76 fl??? «5?»! #0 7511/7 mxfl (MP/D'Q’V’VVV/lléfl/ wfl/ .\ 71E +4? /”’/’/’ e77 gain/)9)! _. 4.0 fivzl/j—bckW/ (Mi)th >43}? , 44/ 9,3; c»;-.,g,@ fir} Ig/ngf £9.23? fluéwwyt k gflég 7/(5 f2”) (M1 Ijrv‘ic'wifé {1‘ fikaf Gui; Va) (010/) 074161” {IN I flWWo/ 50 MM! W5 fl'*/,)/EI3/ MW- 4“ (b) [4 marks] Give an isomorphism from the graph H below to the graph A2. U(llitq) ha {1,7, ?, w} 1,2,; 'r/Izib/ ZIg/V {,3,‘/ l Figure 1: Graph H . x: , E J fly) (twfltv) (1,2,!) /(6,% (any) {(1,3) *2) m. l 1,0,3? 3:012]? IWATH 23.9 Mid—Term Exam, Jun. 27, 2006 10 (C) [4 marks] Is An bipartite? Justify your answer. Ah [)7 $397796- / 1’ 7Z7 0 5451/5? W / 4&1' g iii/1 a 6M“ W: Kr 4/274. )[I/llfvf malf/ é"? f“? i m. % ~ é/MM/ 27/ KM M/y 4 W2 29% 4432/4 49/ ( mi (/7 WW ffl’? OW” flpfivfi /flg/{¢4kf M n , .é/fi’w (a ,4 W i241" 454 fi, 677’ flaw m «WM/éfl‘Ygfé/é ” fl 1’ . 5‘92 5' 1% ‘éi é"? “ "7 I” «(36% MM w»: 74%? 5% 6/} A /%M%‘ W/ g” MW [M 74’” "Wymsflflmm / Mr? ffliiig ,rzfivf/m 4,; (gay/5; Vf/Wf/é‘f M (W// W-M/ Era/“fir ,xM 4f 6?”)! gfiég/fl WM fiaa/z ///g fly; fly” 7977324 [ 4 5/}W/Agbqf ...
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