Department of Mathematics
University of Notre Dame
MATH 10110 - Principles of Finite Math
Spring 2017
Name
Practice Exam 1
February 15, 2017
This exam has 10 problems worth a total of 100 points. You have the entire 50 minutes to work on
it. For full credit, you must show all work. Please write your name in the space provided.
Please read and sign the Honor Pledge:
Honor Pledge: As a member of the Notre Dame community, I will not participate in or tolerate
academic dishonesty.
Signature:
Question
Points
1
10
2
10
3
5
4
8
5
8
6
10
7
8
8
10
9
11
10
12
11
6
Total:
98
Score
1. (a) (2 points) Explain the difference between a permutation and combination.
(b) (2 points) State the (Generalized) Multiplication Principle.
(c) (3 points) What are DeMorgan’s Laws? (Hint: There are two.)
(d) (3 points) Use a Venn Diagram to show one of DeMorgan’s Laws.
2. The following table gives some data about members a family. An experiment consists of choosing
one of the people from this list.
Name
Bob
Mary
Rae
Billy
Neil
Ann
Jim
Hometown
Kansas City
Apex
Apex
Orlando
Kansas City
Alpena
Evansville
Age
33
22
24
50
28
54
58
(a) (2 points) Let A = {People under 30}. List the elements in A.
(b) (2 points) Let B = {People from Kansas City}. List the elements in B.
(c) (4 points) List the elements in A ∩ B. Give a verbal description of A ∩ B. (DO NOT just
say "A intersect B")
(d) (2 points) Let C = {People from Apex}. List the elements in (A ∩ B) ∪ C
3. (a) (2 points) Given that the set A has 8 elements, the set B has 18 elements, and A ∩ B has 5
elements, how many elements does A ∪ B have?
(b) (3 points) Given that S 0 = {a, q, m, s, u} and T 0 = {b, a, u, m}, list the elements of (S ∪ T )0
4. (a) Shade the indicated subset
i. (2 points) A ∪ A0
ii. (2 points) (A ∪ B)0
iii. (2 points) A ∩ C
iv. (2 points) A ∪ (B ∩ C)0
5. In the Eurozone license plates consist of two letters, a dash, three digits, a dash, then two more
letters i.e. AB-123-CD.
(a) (3 points) How many such license plates are possible?
(b) (3 points) If the Eurozone were to exclude three sixes for the three digits, how many license
plates would be possible?
(c) (2 points) If the Eurozone were to include the two letter ISO code of the owner’s nation
before the first two letters of a plate and there are 19 members of the Eurozone, how many
license plates are possible? For example France’s ISO code is ’FR’, so a possible license plate
from France would be ’FR-AB-123-CD’. Ignore part b for this problem.
6. Write the following as a SINGLE multinomial or binomial coefficient.
(a) (2 points) What is the coefficient of x6 y 2 in the expansion of (x + y)8 ?
(b) (2 points) The number of ways to choose a five courses to take out of of 120 offered.
(c) (2 points) The number of 11 letter words (strings of letters) which consist of one M, four I’s,
four S’s and two P’s.
(d) (2 points) The number of unsorted poker hands. (A poker hand consists of 5 cards from a
standard deck of 52).
(e) (2 points) Of nine contest winners, three will receive cars, three will receive flatscreen TV’s,
and three will receive iPads. In how many ways can the prizes be awarded?
7. Consider the set A = {a, b, c, d}.
(a) (4 points) List all subsets of A with two elements. There are 6.
(b) (4 points) List all ordered partitions of A of type (2, 2).
8. (a) (2 points) "Project Fresh" is a Notre Dame club which aims to educate students about breakdancing. If there are 16 members in the club, in how many ways can they select a President,
Vice President, and Treasurer?
(b) (2 points) Out of a litter of 8 puppies, your family plans to buy 3 dogs. How many ways can
you choose the 3 dogs?
(c) (2 points) Given that C(n, 3) = 680 and 3! = 6. Find P (n, 3).
(d) (3 points) How many poker hands consist of 3 Jacks and 2 Queens? Recall a poker hand
consists of exactly 5 cards.
(e) (3 points) How many poker hands consist of 3 hearts and 2 cards from a different suit?
9. Some types of trees, including the giant Redwoods, have fireproof bark. Of the 200 types of trees
in the United States, 20 have fireproof bark, 50 are classified as pine trees and 5 trees are pine
trees and have fireproof bark.
(a) (5 points) Draw a Venn diagram representing this situation. Fill in all appropriate numbers
and make sure to clearly label your diagram.
(b) (2 points) How many trees are neither fireproof nor pine trees?
(c) (2 points) How many trees have fireproof bark but are not pine trees?
(d) (2 points) How many trees are pine trees OR trees with fireproof bark?
10. (a) (3 points) A class of 10 students is going to play poker. The teacher will divide the class into
groups. One group will have 4 students and 2 groups will have 3 students each. In how many
ways can the class be divided into these groups?
(b) (3 points) At a summer camp, 20 children will divide up to do different activities. 10 children
will play capture the flag, 4 children will go canoeing and 6 children will color. In how many
ways can the children be divided into these groups?
(c) (3 points) At a national presidential convention, party rules require that a committee must
consist of 3 men and two women. There are 9 men and 6 women to choose from. In how
many ways can the committee be formed?
(d) (3 points) How many poker hands consist of at least 3 Aces?
11. The following represents a map through a city. The lines represent streets (there are 6 streets
running north/south and 4 streets running east/west). All questions following refer to "shortest
routes" from A to B. That is, only moves south and east are allowed.
A
C
B
(a) (3 points) How many "shortest routes" are there from A to B?
(b) (3 points) How many "shortest routes" do not have two consecutive souths?