Discrete Math 2 Final Exam Review
Name ___________________________
1. A computer system requires a user to have an
access code that consists of a four-digit
number that is not allowed to start with zero
and cannot repeat digits. How many such
codes are possible?
7. Determine the 3rd term in the expansion
of (y - 3)7
8. Expand ( x  3)5
2. Vermont license plates are designed so there
are two letters followed by four digits. How
many different license plates are possible?
3. A student council with 25 members is electing
the following three officers: president, vicepresident, and treasurer. If all members of the
council are eligible for all offices, and no
member can hold more than one office, how
many different slates of officers are possible?
4. In how many ways can a committee of 2 men
and 3 women be formed from a group of 12
men and 8 women?
9. Use the given table to answer the questions.
Male
Female
15
9
13
12
9
13
Rock
Paper
Scissors
P(Male or Rock)
5. Mr. Soto needs five students to help with a
science experiment. In how many different
ways can Mr. Soto select a group of five
students from his class of 28?
P(Paper and Female)
P(Rock│Female)
P(Male | Scissors)
6. In how many distinct ways can the letters of
the word SLEEPLESS be arranged?
10. A bag contains 8 red skittles 5 blue skittles,
and 7 green skittles. Three skittles are drawn
at random and not replaced. What is the
probability that the first skittle is red, the
second one is red, and the third one is blue?
11. In a class of 32 seniors, 18 have a world
language class, and 25 have a science class.
Assuming all of the seniors have either a world
language class or science class, what is the
probability that a class member chosen at
random has a world language class and a
science class?
14. It was determined that only 3% of all adults
have a rare disease. A new test to detect the
disease was developed. The new test properly
identified 99% of the adults with the disease as
having the disease (positive test result). It also
properly identified 96% of the adults that do
not have the disease as not having the disease
(negative test result). The tree diagram below
represents this problem.
Has the Disease
Test
Positive (.99)
Yes (.03)
Negative (.01)
Positive (.04)
No (.97)
Negative (.96)
12. In a game a spinner will land on red 55% of
the time, yellow 15% of the time, and blue
30% of the time. If the spinner is spun three
times, find the following probabilities.
None of the three spins are blue.
What is the probability an adult will have a
positive test? (round to four decimal places)
If an adult is identified as having the disease
(positive test) what is the probability that the adult
actually has the disease? (round to four decimal
places)
The third spin is the first one that is yellow.
At least one spin is red.
13. The Red Line bus is late 18% of the time. If
Ricky plans to ride the train for 20 days, what
is the probability that the train will be late 4 or
fewer times?
15. Raul can hit a target on 38% of his attempts.
Assuming each attempt is independent, what is
the probability that he hits the target exactly 5
times in his next 12 attempts?
16. A new medication is tested and determined to
be effective in 82% of the cases. If 20 people
are given the medication, find the probability
that the medication is effective on 16 or more
people.
Discrete Math 2 Final Exam Review
17. The probability of winning on a slot machine
is 0.15. What are the odds against winning?
What are the odds in favor of winning?
18. What are the odds in favor of spinning a 3?
What are the odds against spinning a 1?
Name ___________________________
Discrete Math 2 Final Exam Review
Name ___________________________
19. Write the following set in roster form:
23. The 36 students in a fine arts class were asked
what activity they participated in. The results
were then converted into the following Venn
diagram.
C   x x  N and 6 < x < 12
20. Use the given information to determine if the
following statements are true or false.
U:
A:
B:
C:
D:
E:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
{1, 3, 5, 7, 9}
{2, 4, 6, 8}
{3, 5, 7}
{2, 5, 7, 10}
{3, 5, 7}
A.
CA
B.
E C
C.
{}  A
D.
{10}  D
E.
CE
21. Find A  B'
How many students were in Band or Choir but not
Theater?
How many students were in Choir and Theater but
not Band?
How many students were in one activity only?
24. Construct a Venn diagram and place the
elements in the proper region.
22. Use the given information to determine A  C '
U:
A:
B:
C:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
{2, 4, 6, 8, 10}
{1, 3, 5, 7, 9}
{5, 6, 7, 8}
U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A: {1, 4, 5, 8, 9}
B: {1, 2, 5, 6, 7, 9}
Discrete Math 2 Final Exam Review
Name ___________________________
25. Does the following have an Euler Path, an
Euler Circuit, or neither?
28. The weighted graph below gives the time in
hours for traveling around five locations.
Using the Nearest-Neighbor method
beginning at the start, what is the minimum
time to complete a Hamilton circuit?
26. Does the following have an Euler Path, an
Euler Circuit, or neither?
27. What is the fewest number of edges that need
to be added to Eulerize the graph?
29. Draw a weighted graph for the mileage
between the Wisconsin cities you plan to visit.
If your plane lands in Madison, what is the
minimum distance traveled to visit each city
and return to Madison? (check all options)
Kohler
Madison
Oshkosh
Wausau
Kohler
-
110
87
160
Madison
110
-
86
143
Oshkosh
87
86
-
104
Wausau
160
143
104
-
Discrete Math 2 Final Exam Review
Name ___________________________
30. Determine if each path name below describes
a Hamilton circuit for the graph? (There is
more than one.)
33.
abcdefg
abdefgca
acdefgba
abdefgdca
abefgdca
dbacgfed
31. Represent each vertex-edge graph with an
adjacency matrix.
Represent each adjacency matrix with a
vertex-edge graph.
34. Given the adjacency matrix below, how many
two-step paths are there from B to D in the
digraph it represents?
M=
A
B
C
D
A 0
B  2
C 1

D 0
1
1
0
1
1
0
1
1
1
1 
1

0
35. Using Kruskal’s Algorithm what is the
minimum cost for the weighted graph below?
A
C
B
D
E
Discrete Math 2 Final Exam Review
Name ___________________________
36. The following graph shows the flight mileage
of Jet One Airline. What is the shortest time to
get from Boston to LAX based on this
spanning tree?
39. Which graph has a larger chromatic number?
37. Which subgraph is a spanning tree of the
graph
38. What is the chromatic number for the
following graph?
40. The director of a dog rescue shelter is trying
to minimize the number of cages used for a
new group of rescued dogs. Certain dogs have
been identified as not being compatible within
a cage, as they may fight or injure each other.
The table below shows the incompatibilities
between the dogs, in the sense that an X
indicates that it is unwise to allow those dogs
in the row and column that meet at the X to be
in the same cage. What is the minimum
number of enclosures needed to minimize
conflict?
Doodle
Doodle
Duke
Fang
X
X
KoKo
X
Duke
X
X
Fang
X
X
X
KoKo
Ruff
X
X
X
Ruff
X