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Unformatted text preview: ll [SOLUTION Math 125002 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 5digit natural numbers have
the sum of digits equal to 3? Stee‘ﬁ‘tA‘ CL {:5 chad W mam66A“ VEH‘WS‘W
m‘lMpreﬂtak “—5 F‘d‘hkj {MARC «@115 la 3‘4..th ‘TLL MUS—q— ozf W33 is. (Sea qEimdrwq 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?
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CH ”Una. Mindakvok 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 (0625— “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
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l HHDODt—‘I—OD 1
1
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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 45 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 consistof m and k vertices? mK (b) A complete bipartite graph G has 20 edges. What is the number
of vertices of 9 if g is Eulerian? (a) 20:3420 _iL\ WLQOLSLs («DMD/{(17) ML 2 240 Mas we! vwﬁuaso
a") 20 Wm Ekmiw‘ (C) 20 : A}. 5 Eek \Hv— consc. (‘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/(aeb {kc/$4.041! 51c; A,B/C MS?!”
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IkQJﬂ/wk Ma‘s 3w‘cii/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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