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Final Term Exam for Applied Combinatory | MATH 3012, Exams of Mathematics

Material Type: Exam; Professor: Trotter; Class: Applied Combinatorics; Subject: Mathematics; University: Georgia Institute of Technology-Main Campus; Term: Fall 2003;

Typology: Exams

2010/2011

Uploaded on 06/01/2011

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Student Name and ID Number
MATH 3012 Final Exam, December 8, 2003, WTT
1. Consider the following seven letter alphabet: {A, B , C, D, E, F , G}.
a. How many strings of length 19 have exactly 6 B’s?
b. How many strings of length 19 contain exactly 4 A’s, 2 B’s, 7 D’s and 6 G’s (and no C’s, E’s
or F’s)?
c. Of the strings described in part b, how many have all 4 A’s before the 7 D’s?
d. Of the strings described in part c, how many have the 4 A’s and the 7 D’s occuring together
as a block of 11 consecutive characters?
2. How many integer value solutions to the following equations and inequalities:
a. x1+x2+x3= 59, all xi>0.
b. x1+x2+x3<59, all xi>0.
c. x1+x2+x3= 59, all xi0.
d. x1+x2+x359, all xi0.
e. x1+x2+x3= 59, all xi>0, x3>12.
3. In three space, consider moves from one point with integer coordinates to another formed by
adding one of (1,0,0), (0,1,0) and (0,0,1). How many paths from (0,0,0) to (8,5,9) can formed
with such moves?
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Student Name and ID Number MATH 3012 Final Exam, December 8, 2003, WTT

  1. Consider the following seven letter alphabet: {A, B, C, D, E, F, G}.

a. How many strings of length 19 have exactly 6 B’s?

b. How many strings of length 19 contain exactly 4 A’s, 2 B’s, 7 D’s and 6 G’s (and no C’s, E’s or F’s)?

c. Of the strings described in part b, how many have all 4 A’s before the 7 D’s?

d. Of the strings described in part c, how many have the 4 A’s and the 7 D’s occuring together as a block of 11 consecutive characters?

  1. How many integer value solutions to the following equations and inequalities: a. x 1 + x 2 + x 3 = 59, all xi > 0.

b. x 1 + x 2 + x 3 < 59, all xi > 0.

c. x 1 + x 2 + x 3 = 59, all xi ≥ 0.

d. x 1 + x 2 + x 3 ≤ 59, all xi ≥ 0.

e. x 1 + x 2 + x 3 = 59, all xi > 0, x 3 > 12.

  1. In three space, consider moves from one point with integer coordinates to another formed by adding one of (1, 0 , 0), (0, 1 , 0) and (0, 0 , 1). How many paths from (0, 0 , 0) to (8, 5 , 9) can formed with such moves?
  1. Use the Euclidean algorithm to find d = gcd(780, 2772).

Then find integers x and y so that d = 780x + 2772y.

  1. Consider the partially ordered set (poset) shown below:

20

8

17

1

5

10

7

15 (^1216)

11

9

2

19

13

14

23

(^3 )

18 21

6

24

25

a. Find a minimum partition of this poset into antichains.

  1. Show that G is hamiltonian by listing the vertices in an appropriate order, starting with vertex 1.

1 2

3 4

5

6

7

8

9

10

  1. Explain why the following graph does not have an Euler circuit. Then show that all the edges can be traced exactly once starting at one vertex and ending at another by listing the vertices (with repetition allowed) in an appropriate order.

1 2 3

4 6

7

5

(^8 )

  1. Show that G is an interval graph and has a representation in which the order of the left endpoints matches the labels on the vertices. Then color the graph using First Fit and find the chromatic number χ(G) and the maximum clique size ω(G).

1

2

3 4

6

7

5

  1. Draw a set of 8 intervals and label them with the numbers from { 1 , 2 ,... , 8 } so that (a) the associated interval graph has maximum clique size 2; and (b) if the vertices of the graph are colored using First Fit in the order they have been labelled, then 4 colors will be used.
  2. Verify Euler’s formula for the following planar graph.

1 2

3

6

5

4

  1. Suppose G is a graph with 80 vertices and 1837 edges. Explain why G contains a triangle.
  2. Suppose G is a graph with 100 vertices and 386 edges. Explain why G is non-planar.
  3. Consider the graph G with non-negative edge weights given in the following text file shown below on the left (note that the file has been sorted by edge weights). In the space to the right List the edges in order which result from applying Kruskal’s algorithm (avoid cycles) to find a minimum weight spanning tree of G. Then set vertex 1 to be the root and list in order the edges which result from applying Prim’s algorithm (build tree).

graph.txt Kruskal Prim 3 5 14 4 7 18 5 6 19 3 6 22 1 2 29 2 4 32 1 5 38 2 6 39 3 7 40

  1. The matrix given (below left) is the distance matrix for a digraph D whose vertex set is { 1 , 2 , 3 , 4 , 5 , 6 , 7 }. The entry di,j denotes the length of the directed edge from vertex i to vertex j. Edges not present in the graph have infinite length. In the space to the right, apply Dijkstra’s algorithm to find all the shortest paths from vertex 1 to all other vertices in the digraph D.

D 1 2 3 4 5 6 7

  1. Extra Credit Consider the positive integers p 1 and p 2 given below (they each have more than 200 digits so it takes several lines to display them): p 1 = 6438080068035544392301298549614926991513861075340 134329180734395241382648423706300613697153947391340909229373325903 847203971333359695492563226209790366866332139039529661751070967691 80017646161851573147596390153

p 2 = 5612568098175228233349808831356893505138383383859 489982166463178457733717119362424318136005466967841045532911243455 294271708400354138459486412994014504308676003129248334006892350611 5878221189886491132772739661669044958531131327771

Both p 1 and p 2 are primes. Now consider the number n = p 1 p 2 formed by multiplying them together. Two students, Alice and Bob, are asked to find φ(n), where φ is the Euler φ function. But Alice is given p 1 and p 2 while Bob is just given the value of n. Why does Alice have an unfair advantage over Bob? Does it help Bob to be told that n is in fact the product of two primes—without being told the values of these two primes?

  1. Extra Credit Show that a tournament always has a directed hamiltonian path.