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exam3solution - ll[SOLUTION Math 125-002 Fall 2009 Exam 3...

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Unformatted text preview: ll [SOLUTION Math 125-002 Fall 2009 Exam 3 Name: Last... First... Book closed. Calculator and notes may not be used. YOur answers to problems 1, 2, 3 may contain expressions like 13!, P(7,3), ( g ), DH, etc. YOu do not have to finish arithmetics. (1) (15 points) How many different 5 5-digit natural numbers have the sum of digits equal to 3? Stee‘fi‘tA‘ CL {:5- chad W mam-66A“ VEH‘WS‘W m‘lM-prefltak “—5 F‘d‘hkj {MARC «@115 la 3‘4..th ‘TLL MUS—q— ozf W33 is. (Sea q-E-imdrwq gens-£124.; an W45?“ \""k‘"\ (2) (15 points) What is the number of permutations of the letters in the word QUESTION such that... .. no letter remains in the original position? :D 3 "exactly two letters remain in their original positions? ( a). D .. exactly six letters remain in their original positions? (a). )1: (8 ”>20 exactly seven letters remain in their original positions? 0 (i; Sexton eai‘i‘brs Fem-ts C\ W £136qu pas HEM-C “Ham ‘H‘L remix“) 8 *“ 4&an Va Maw-L5 CH ”Una. Mindak-vok Vagina/H 0.50/06" Aflmdwe Sean” or PM 4— {I / (3) (15 points) Find the coefficient (a) of :37, (b) of is, (c) of t9 in the Q binomial expansion of (2t2 + as. 1 1‘ 2‘1 t2 (2{1+ éfvé _____ (2*?)9 + (%)(24c )S-(-‘E)+(%_) (2%)(¥)+\L. h c, w— '5 q i» .“. é 0'“ W .. 2 ‘t + 69. 't + (2)2 JC +‘”' pit—Eekfi'fr" 4%?“ Aksww‘. (at) O (QC (06-25— “3' I (4) (10 points) A graph G has adjacency matrix DHOHHOOO l—‘OHHDDDH ODDHHDl—‘O OOHt—‘Dl—‘OO 1 0 1 1 0 0 0 l HHDODt—‘I—OD 1 1 0 0 0 1 1 0 OCHOHHOD What is the number of vertices of Q? 8 ( +L_,_ Mam/6M 0:6 ants) ais e‘nu eroe 3 {TM W {—5 Wht th rub fdge ofg? IH (do/VLW jfiqgfld What is the degree sequence of 9? (f! (f 2 (1 j '1 J 3/ 3/ 3/ 3 (mm W 4% of 4-5 ta W4 3M W 13:4“ {a M“ {amazij W3 (5) (15 points) (a) What is the number of edges in the compiete bi— partite graph whose parts consist-of m and k vertices? m-K (b) A complete bipartite graph G has 20 edges. What is the number of vertices of 9 if g is Eulerian? (a) 20:34-20 _iL\ WLQOLS-Ls («DMD/{(17) ML 2 2-40 Mas we! vwfiuaso a") 20 Wm Ekmiw‘ (C) 20 : A}. 5 Eek \Hv— cons-c. (‘2)(W W is Wear MA) 41% vu‘i'ic—Ls awe MW) 50 “WEE ‘ Aksvw ‘. 2H0 3: \2 (6) (15 points) Is the graph shown in the picture Hamiltonian? IEYES, describe a Hamiltonian cycle. If NO, prove why not. Vflf‘kte" A}B)0‘mo‘(c have Mafia 2“; So 504% ee/(a-eb {kc/$4.041! 51c; A,B/C MS?!” & {MAC/Lax a w memk 95:11 (:4 m “3345) ka ‘HI-an ‘HL VMJW)‘ O “(j/“04¢; MEW WM Ms'h ’69. K. ‘- IkQJfl/wk Ma‘s 3w‘c-ii/L 15 INF-$5: m . hm firm/FL a“. Eff. Haméfinim (7) (15 points) Are the digraphs represented in the two pictures iso— morphic? If YES, Show an isomorphism by labelling vertices. If N0, prove why not. ...
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