This is a custom exam written by Trevor, from TrevTutor.com, that covers Counting,. Catalan Numbers, Discrete Probabilit
Discrete Mathematics 2
Name:
TrevTutor.com Midterm 1
Class Section
Time Limit: 60 Minutes
This exam contains 7 pages (including this cover page) and 6 questions. The total number of points is 50. This is a custom exam written by Trevor, from TrevTutor.com, that covers Counting, Catalan Numbers, Discrete Probability, and the Inclusion-Exclusion Principle. A lot of these questions are fairly straight-forward and are designed to show that the basic concepts and ideas of each section are understood. Most questions are variations of the examples shown in videos. One question will rely on material taught in Discrete Mathematics 1. If you would like to see more practice exams, as well as the solution to this exam, check out TrevTutor.com. Question Points Score 1
10
2
8
3
8
4
10
5
10
6
4
Total:
50
Discrete Mathematics 2
Midterm 1 - Page 2 of 7
1. (10 points) The following questions are related to counting. (a) (2 points) How many ways can we distribute 13 plants among 6 di erent people?
(b) (2 points) How many binary strings of length 10 contain exactly 4 zeros?
(c) (2 points) How many positive integer solutions are there to the equation x1 + x2 + x3 + x4 = 20 where x1 4, x2 6, and x3 ; x4 0?
(d) (4 points) How many ways can Dr. Martin arrange 20 textbooks on three shelfs so that each shelf as at least 3 books, but no more than 9?
Discrete Mathematics 2
Midterm 1 - Page 3 of 7
2. (8 points) Write out the Axioms of Probability, then prove that P (A) = 1 those axioms. Axiom 1: Axiom 2: Axiom 3:
( ) using
P A
Discrete Mathematics 2
Midterm 1 - Page 4 of 7
3. (8 points) The following questions are more challenging counting questions. (a) (2 points) What algebraic product corresponds to the path E E E N N E N E N N E N ?
(b) (2 points) How many ways are there to parenthesize abcdef ghij ?
(c) (2 points) Prove that 1
2n + 4
n 2
8
n 3
+ + ( 2)n
n n
= ( 1)n
(d) (2 points) Six students write a Discrete Mathematics 1 exam. Unfortunately, the markers for the exam have gone on strike because the department will no longer give them 4-ply toilet paper. How many ways can we redistribute the six exams so that no student marks his own?
Discrete Mathematics 2
Midterm 1 - Page 5 of 7
4. (10 points) The following questions relate to probability. (a) (2 points) If P (A) = 0:4; P (B ) = 0:6, and P (A \ B ) = 0:7, what is P (A [ B )?
(b) (4 points) In Discretability Land, 20% of residents own a philosophy textbook and a linguistics textbook. Also, 70% of residents that own a philosophy textbook also own a linguistics textbook. What percentage of residents do not own a philosophy textbook?
(c) (4 points) Suppose that 15% of people are cannibals, and that the rest are not. Also, suppose that 40% of cannibals like rabbit meat, but only 25% of non-cannibals like rabbit meat. If you see a man eating rabbit meat, what is the probability that he is a cannibal?
Discrete Mathematics 2
Midterm 1 - Page 6 of 7
5. (10 points) Suppose that you roll two fair dice to get a pair (x; y ), where x and y belong to f1; 2; 3; 4; 5; 6g. Event A is that x = y , and Event B is that x + y is a multiple of three. (a) (3 points) Determine P (A), P (B ), and P (A \ B ).
(b) (2 points) Are Events A and B independent?
(c) (2 points) Are Events A and B mutually exclusive?
(d) (3 points) Suppose Event C is that x + y = 4. Determine each combination of events to be either mutually exclusive or not.
Discrete Mathematics 2
Midterm 1 - Page 7 of 7
6. (4 points) Markiplur University has recently hired 5 janitors after noticing that toilets were consistently being clogged by 4-ply toilet paper. There are 23 toilets that are currently over owing. How many ways can we assign the janitors to the toilets so that exactly two janitors have no work to do?
