# University of Illinois at Urbana Champaign Spring

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Niveau: Supérieur, Doctorat, Bac+8
University of Illinois at Urbana-Champaign Spring 2007 Math 181 Group F1 Midterm 2. Correction. 1. The table below shows chemical compounds which cannot be mixed without causing dangerous reactions. Draw the graph that would be used to facilitate the choice of disposal containers for these compounds ; what is the minimal number of containers needed ? A B C D E F A B C ED F X X X X X X X X X X X X X X X X D 3 C E 2 AB 1 F 1 2 1 Answer. The graph is above ; a vertex-coloring of this graph will at least use three colors since the graph contains a triangle, and the coloring above shows that 3 colors is enough. So the chromatic number of the graph is 3, which means that the minimal number of containers needed is 3. 2. (a) In designing a security system for its accounts, a bank asks each customer to choose a ﬁve-digit number, all the digits to be distinct and nonzero. How many choices can a customer make ? Answer. There are 9? 8? 7? 6? 5 = 15120 possible choices. (b) A restaurant o?ers 4 soups, 10 entrees and 8 desserts.

• fifty voters

• voters' preference

• voting system

• voters

• system satisﬁes

• vertex-coloring using

• most approval

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2
E
A B C D E F
A X X X AB
13
B X X X X
C X X
D X X X
21
C DE X X
F X X
F
1
3
3 3
9×8×7×6×5 = 15120
4 10 8
3
4× 10× 8 = 320
5 4×10×5 = 200
50
(50×49)/2 = 1225
32× = 62
13
4 5
2 3
4 5= 6 = 102 3
1
1 1
1 2 1 4C
21 3 3 1 5C
1 4 6 4 1 3
1 5 10 10 5 1
5
52 = 32 5
12 8 6 3
First choice B C A C
Second choice C A B B
Third choice A B C A
8×1+6×2 = 20 12×2+6×1+3×1 = 33
12×1+8×2+3×2 = 34
18
C 11
Number of voters
20 14 10 6
First choice A B B C
Second choice C A A D
Third choice E D C B
Fourth choice B C D A
Fifth choice D E E E
50 0 44 6 D 44 6 30 20 26
24
20−30 24 26 36−14 50−0
A X X X X X X X
B X X X X X
C X X X X X
D X X X X X X
87.5 62.5 75
A A C C B
B B A A C
C C B B A
C
B C
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