Discrete Maths
Which is a correct logical conjunction formed with the following phrases? Drew likes roses. Drew loves lilies.
Drew likes roses and Drew loves lilies.
Which one is the correct denotation of an empty set?
E = { }
Which rule represents the nth term in the sequence 9, 16, 23, 30...?
an = 7n + 2
If U = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and A = {12, 13, 14, 20}, what is the complement of A?
{11, 15, 16, 17, 18, 19}
In the sequence 9, 14, 19, 24, 29,.... Say we're using a to describe the terms. What is the value of the term below? a3
19
Select the appropriate truth table for the tautology p --> (p v q) .
4 lines if truth is correct answer
Which sequence below is a geometric sequence?
5, 15, 45, 135, 405, ...
Express this series using sigma notation: 3 + 5 + 9 + 17 + 33.
E5 2n+1
What kind of reasoning does the statement below illustrate? My phone starts to ring whenever my cat starts to purr.
False cause
Select the tautology from the statements below.
I either order pizza or I don't order pizza.
What is the rule for the nth term of the sequence with a7 = 53 and a13 = 101?
an = 8n - 3
The shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 180 degrees in the counterclockwise direction. What are the coordinates for A'B'C?
(-1,-1), (-4,0), (1,1)
The shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 90 degrees in the counterclockwise direction. What are the coordinates for A'B'C?
(-1,1), (0,4), (1,-1)
Which is the correct order for the steps to find a solution of a homogeneous linear recurrence?
(1) find the characteristic equation (2) find the roots of the characteristic equation (3) compute the solution coefficients
A statement that is True regardless of the truth values of all the variables in it is called a tautology. Choose the tautology from the list below.
(NOT q) OR q
Let m stand for 'Math is fun', n stand for 'Today is Thursday', and p stand for 'It is raining.' Write a Boolean expression for the statement 'Today is Thursday and it's not raining, or math is fun.'
(n N-p)V m
You play a game with two six-sided dice. If you roll a sum of 6 or 8, you win $3. If you roll a sum of 11, you win $1, but for anything else, you lose $2. If you continue to play this game, what do you expect to win in the long run?
-$0.44 per roll
Suppose you play a game with two six-sided dice, where if you get doubles, you win $10, but for anything else you lose $5. If you continue to play this game, how much do you expect to win or lose per roll in the long run? In other words, what is the expected value of this game?
-$2.50
Naomi is buying a ticket for the lottery. She can pick 5 numbers, 1 through 15, with no numbers repeating. Order does not matter for her to win. What are her chances of winning?
1 in 3,003
Hannah is playing the lottery. She can pick 5 numbers between 1 and 32, with no numbers repeating. The order of these numbers does not matter. She can also pick a bonus number between 1 and 20. What are her chances of picking all of the numbers, including the bonus, correctly?
1 in 4,027,520
What is the chance of rolling a 3 on any given six-sided die?
1 in 6
Which vertices can be your starting point? 45123
1 or 3
Which truth table output corresponds to the following circuit? Tuppo vako ani tuppo navako
1, 0, 0, 1
Calculate the first 5 terms in this recursive function.
1, 2, 4, 8, 16
Find a Hamilton path.12543
1, 3, 4, 5, 2
Which of the following sequences is NOT a geometric sequence?
1, 8, 15, 22, 29, ...
In Mr. Martin's math class there are 15 students out of which 4 will be selected to receive a special award. How many combinations of students can be selected?
1,365
Knowing the sequences A and B and their corresponding generation functions, which is the generation function for sequence C?
1/1-x2
Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that girls will place 1st, 2nd, and 3rd in the marathon?
1/114
Annie writes the numbers 1 through 10 on note cards. She flips the cards over so she cannot see the number and selects three cards from the stack. What is the probability that she has selected the cards numbered 1, 2, and 3?
1/120
At a local ice cream parlor, there are 6 different flavors of ice cream from which to select (strawberry, chocolate, vanilla, rocky road, mint chocolate chip, and peanut butter). Each ice cream sundae must include 2 different flavors of ice cream; what is the probability that the next customer's sundae will have vanilla and chocolate (in either order)?
1/15
Jimmy has the letters for the state of MISSISSIPPI written on cards, one letter per card. He turns the cards over and mixes up the order. If he selects one card at a time without replacing the cards, what is the probability that he will spell the word MISS in order?
1/165
When you roll two six-sided dice, what is the probability of getting a sum of 11?
1/18
A newspaper company is selecting four houses to receive a free newspaper on your block. There are 10 houses on your block that are numbered 1-10. What is the probability that the four houses selected for a free newspaper will all be even numbered houses?
1/42
When two six-sided dice are rolled, what is the probability of getting doubles? (two ones, two twos, etc.)
1/6
The total number of degrees in a graph is 20. How many edges does it have?
10
Calculate the first 5 terms of this recursive function.
100, 50, 25, 12.5, 6.25
A woman is deciding what to wear to work. She is considering an outfit from among 5 blouses, 3 skirts, and 7 pairs of shoes. How many possible outfits are there?
105
Six people are going to sit at a round table. How many different ways can this be done?
120
Solve the expression 5P4 (P = permutation )
120
Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, ...
128(1/2)n-1
Jon and Alex are playing an arcade game. Jon is pretty good at this game. He has an 80% chance at winning this game. He and Alex will both play 8 games. Jon will need to win 5 games to beat Alex. What is the probability that Jon will win exactly 5 games?
15%
Jon and Alex play a game where they each need to shoot a puck into a net. They are equally matched (thus each has a 50% chance of getting the puck in the net) and will each shoot 12 pucks. What is the probability that Jon will score exactly 7 times?
19
Where does Boolean logic derive its name from?
19th century mathematician George Boole
A complete graph with 3 vertices has how many Hamilton circuits?
2
A graph with an Euler path can have at most how many odd vertices?
2
How many steps are in mathematical induction?
2
Jon and Alex play a game where they each need to kick a ball into a net. They will each kick 11 balls, but Alex has a slight edge, with a 60% chance of scoring to Jon's 40%. What is the probability that Jon will score exactly 8 times?
2
Which number needs to be changed to find the inverse of this matrix? [100 012 001]
2
Which of the following circles would be considered a different arrangement?
2
Perform the first step of mathematical induction for the mathematical statement n + 1 > n.
2 > 1
Which of the following is an Euler path for this graph? 12cross 3 4
2, 3, 4, 2, 1, 4
Calculate the first 3 terms of this recursive function.
2, 4, 6
Jon and Alex decide to play skee ball. This game only has one ring, and the guys must roll the ball into the 10,000 point ring or they won't get any points. Each of them has a 50% chance of success with each roll and will each roll 6 balls. What is the probability that Jon will get the ball in the ring exactly 4 times?
23
Calculate 4! (factorial)
24
How many rows would a matrix that represents a system of 5 variables and 3 equations have?
3
Suppose that a dating service is determining couples that would make a good match. In the graph shown, two sets of clients are represented with vertices, and the edges between the vertices indicate that the clients make a good match. Based on the graph, how many possible matches are there for Erin?
3
Suppose you have a drawer full of white socks and black socks. What is the minimum number of socks you would need to pull out of the drawer to guarantee a pair of matching socks?
3
Jon and Alex play a game where they shoot water at a target in order to make a car move on a track. The person that gets their car over the finish line first wins. Alex and Jon decide to play the game 3 times. If they are evenly matched, then what is the probability that Jon will win 2 games?
38
When you multiply matrices, if the first matrix is a 4x3 matrix, which of the following matrices can we multiply with the first?
3x5
Solve 8! (the factorial of 8)
40320
Solve the expression 7P2 (P = permutation)
42
Calculate the probability of getting a single pair in a hand of poker. Round that probability to the nearest percent.
42%
Jane is attempting to unlock her locker but has forgotten her locker combination. The lock uses 3 numbers and includes only the numbers 1 to 9. Each of the digits also can not be repeated in the combination. How many possible locker combinations can be formed?
504 locker combinations
Say you're playing Blackjack, and you have a four and a jack. If you see that one of the dealer's two cards is a five, then what's the probability that you will be dealt a card that helps you?
54%
A couple wants to plant some shrubs around a circular walkway. They have seven different shrubs. How many different ways can the shrubs be planted?
720
A horse race includes 10 participants. How many possible finishes are there for the top three positions: first (win), second (place), and third (show)?
720
Evaluate the following: 6En=1 2(3)n-1
728
The local bowling team plays in a 7-team league where each team plays other teams 4 times in a season. Using the combination formula, how many different games will be played in a season?
84
A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). On one spin of the roulette wheel, what is the probability that the ball lands on a red slot?
9/19
The rotation matrix for 270 degrees in the counterclockwise direction is the same one as the one for a:
90 degrees clockwise rotation
Which of the following would be the 14th term of the sequence below? 11, 22, 44, 88, 176, ...
90,112
Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that boys will place 1st, 2nd, and 3rd in the marathon?
91/228
Find the sum of the arithmetic series 106 +100 + 94 + ... + 28 + 22.
960
Given the sequences A and B and their corresponding generating functions, which would be the sequence C that corresponds to the right most generation function indicated below?
< 0, 1, 4, 11, 26, 57, ... >
Which of these sequences corresponds to the generating function below?
< 1 , 2 , 4 , 8 , 16 , ... >
Given the following truth table, which one of the following diagrams represents the simplest possible circuit CBAZ
A arrow BC tuppo navako again tuppo navako tuppo vako z
Which graph in discrete mathematics has a path of edges between every pair of vertices in the graph?
A connected graph
What is the difference between a directed and an undirected graph?
A directed graph uses arrows to indicate one-way relationships, but an undirected graph does not.
What is game theory?
A field of analysis used to guide logical decision-making.
A complete graph is a graph where each vertex is connected to how many of the other vertices?
All
In the graph shown, the vertices represent cities, and the edges represent flights between those cities. The weights of the edges represent the cost of the flights, in hundreds of dollars. Based on this, what two cities are the cheapest to fly between, and what is the cost?
Flight: City A and City E Cost: $200
Suppose you play a game where you spin a spinner (see picture below) with areas of the colors on the spinner broken down as shown: 10% blue, 60% green, and 30% red. In addition, if the spinner lands on red you win 6 points, if it lands on blue you win 1 point, and if it lands on green you lose 5 points. If you keep spinning, how many points can you expect to win or lose per game?
I expect to lose 1.1 points per game.
You play a game where you toss a coin. On each toss if it lands with heads up, you win $1. However, if it lands with tails up, you lose $2. If you continue to play this game, how much can you expect to win or lose per game?
I expect to lose 50 cents per game.
A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). Suppose that for a bet of $1 on red, the casino will pay you $2 if the ball lands on a red slot (a net gain of $1), and otherwise you lose your dollar. What can you expect to win or lose in this game?
I expect to lose about 5 cents per game.
Which of these is the logical contrapositive to the following statement? If killing in any sense is wrong, then murder is wrong.
If murder is not wrong, then killing in any sense is not wrong.
All of the following statements could be proven with a direct proof EXCEPT:
If n is an odd integer, then m is an even integer.
What is the hypothesis in the following statement? 'If p is an even integer and q is an odd integer, then p + q is an odd integer.'
If p is an even integer and q is an odd integer
Which of the following is the correct Karnaugh map for this Boolean expression? Output = AB
Output 0001
Which sequence corresponds to a depth-first search for the following graph? PQRST
P-->Q-->R-->S-->T
Finding the minimum spanning tree by starting at a random node and adding the node with the lowest weight link is called _____.
Prim's Algorithm
A Karnaugh map does what to Boolean expressions?
Simplifies them
Which of the following statements is NOT true?
There are only three types of graphs in discrete mathematics.
Calculate the probability of getting a straight in a hand of poker.
There is approximately a 1 in 255 chance of getting a straight in poker.
What does a computer use to store binary numbers?
Transistor switches in the open or closed position
A _____ is a scenario where every point in a figure is moved the exact same distance and in the same exact direction.
Translation
Which of the following combinations of gates would produce the following truth table? X Y ?
Tuppo vako ani traingle
Suppose we wanted to use mathematical induction to prove that for each natural number n, 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1). What would we show in the base step for n = 1 and n = 2?
We would show that the statement was true for n = 1 and for n = 2 by plugging 1 and 2 into our formula separately, and making sure they both make a true statement.
Knowing the generating function corresponding to the sequence of the natural numbers (except 0) in the first row on the image below, which would be the generating function for the sequence in the second row of the image below?
x2/(1-x)2
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Which of the following is not a subset of the universal set?
{8, 9, 10}
From mortality tables it has been determined that the probability of a 20-year old male non-smoker dying within the year is 0.0035. Suppose an insurance company wants to sell a $50,000 1-year life insurance policy to a 20-year old non-smoking male. What should they charge for the policy to break even?
$175
What is the 83rd term of the sequence 91, 87, 83, 79, ... ( = a1, a2, a3, a4, ...)?
-237
A graph with an Euler circuit can have at most how many odd vertices?
0
The output from 1 (TRUE) AND 0 (FALSE) would be:
0
What is the third term in this recursive function?
0
If the domain of the function f(x) = x^2 is 0 ≤ x, what is the range of f?
0 to infinity
Which of the following is a Euler path? 12034
0, 2, 1, 0, 3, 4
The output of 1 (TRUE) OR 0 (FALSE) would be:
1
A television channel conducts a study on the number of TVs per household in their service area. Out of 1000 households surveyed, 350 have one TV, 500 have two TVs, 120 have three TVs and 30 have four TVs. How many TVs does each household have on average?
1.83
A sandwich shop offers 3 types of meat, 3 types of cheeses, 3 types of bread, and 4 types of condiments for its sandwiches. If you select one of each of the four ingredients, how many possible sandwiches can you create?
108
Consider an example with 100 students where 20 are taking discrete math, 30 are taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking discrete math and web design, 10 are taking Java and web design and 5 are taking all three classes. How many students are taking discrete math only?
11
Which choice below represents an arithmetic series?
12 + 10 + 8 + 6 + 4
10 C7
120
A locker combination contains four numbers between 1 and 20 and none of the numbers can be repeated. What is the probability that the locker combination will consist of all even numbers?
14/323
If 55 and 89 are two sequential terms of the Fibonacci sequence, what term would come next?
144
If a2 = 5 and a8 = 35, what is the value of a30?
145
What is the sum of all the even integers between 2 and 250?
15700
According to Fleury's algorithm, how many odd vertices does a graph with an Euler path in it have?
2
At most, _____ of the verticies will be of an odd degree in a semi-Eulerian graph.
2
For this Karnaugh map, how many inputs give you an output of 1?
2
How many different values can a Boolean represent?
2
How many odd vertices can a graph have in order to use Fleury's algorithm?
2
How many odd vertices does this graph have? 12 cross 34
2
What is the number in the first row and first column in the answer matrix? [424 580][12 20 31]
20
A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. How many bricks are in the 12th row?
27
Evaluate f(x) = √(x), when x = 9.
3
Find the sum of the following geometric series:
3
Which of the following will get you from point 3 to point 5?
3 to 2 to 5
A ball is dropped and begins bouncing. On the first bounce, the ball travels 3 feet. Each consecutive bounce is 1/8 the distance of the previous bounce. What is the total distance that the ball travels? Round to the nearest hundredth.
3.43 feet
Suppose you toss three coins. What is the probability that you get two heads and one tail if the order in which you get them does not matter?
3/8
3. Which element is 'maximal' in the following diagram?
30
If a graph has 15 edges, what must the degrees of the vertices add up to?
30
Asia is conducting an experiment for her psychology class. She asks 80 students if they believe in hypnosis. Previous studies show that 60% of the population believes in hypnosis. Based on the expected value, how many people should answer yes, they do believe in hypnosis, in Asia's study?
48 peeps
Which system of equations is represented by the following matrix? (Think of the vertical line as the equal sign.) [4 -1 1][10 -5 1]
4x-y+z=10, 5x+8y+7z= -5, 6x+8y-2z = 1
How many vertices are in this graph? 645321
6
What is the number in the second row and third column in the answer matrix?[424 580]+[126 207]
7
How many Euler circuits are in this graph? 1234
8
A game of Blackjack is played. Suppose you are the first player dealt with a new pack of cards. What is the probability you get an ace and a jack (in any order)?
8/663
Given item A, which of the following would be the value of item B?
80
For this Boolean expression, which inputs will give you a 1? Output = AB
A = 1 B = 1
Which of the following is not a finite set?
A = The set of all real numbers
Which one of the following pairs of sets is equal?
A = {1, 2, 3} B = {3, 1, 2}
What is a directed graph?
A graph with directed edges
Which of the following is NOT a property of a random variable?
A random variable cannot be negative.
A matrix is:
A rectangular array of numbers, symbols, or functions
What's the difference between a sequence and a series?
A sequence is an ordered list of numbers, and a series is the sum of a sequence's terms.
What is a tautology?
A statement that is always true
The set of all elements that are under consideration for a particular problem or situation is known as:
A universal set
Which of the following is a Hamilton circuit? ABCDEF
A, B, C, D, E, F, A
When you use the advanced settings in an online search engine and use the option to search for 'all these words', which type of Boolean logic are you using?
AND
How many vertices in a graph do you visit by traveling a Hamilton path or circuit?
All of them.
Why would a person who has few wrecks benefit from a lower premium?
Because even though a low premium comes with a high deductible, if the driver has a low chance of a wreck, they're less likely to ever need to pay the deductible.
What do we call the probability of each success in a given binomial experiment?
Binomial distribution
What do we call an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure?
Binomial experiment
How are logic circuits built?
By combining logic gates according to the function expression.
Which of the following circuits is the simplest that corresponds to the following function: Z= abc'+abc ?
CBA ani join to one tuppo navako
What is the brute force method?
Calculate each Hamilton circuit to see which one is best.
Which is the nearest neighbor method?
Choose the next closest city.
What is a probability formula that uses factorials to find the number of possible combinations of all the outcomes in an experiment?
Combination formula
Which statement best describes combinatorics?
Combinatorics is a fancy name for the study of counting.
Eulerize this graph. 12 cross 34
Connect vertices 2 and 4 with another edge.
When one event affects the outcome of another event, we call them:
Dependent events
Trees are diagrams which are part of _____.
Discrete Math
What is data that cannot be divided, is distinct, and can only occur in certain values?
Discrete data
What is the probability related to discrete data?
Discrete probability
Express this series using sigma notation: 5 + 6 + 7 + 8 + 9 + 10...
E infinity 4+n
Why are NAND and NOR gates considered to be functionally complete?
Each could be used exclusively to form any other gate operation.
After finding the number of ways to get four-of-a-kind for four cards in a poker hand, should you use: a. 48 choose 1 or b. 12 choose 1 x 4 choose 1 to describe all possible ways for getting the 5th card? Explain your reasoning.
Either way is correct, because both calculations describe all possible ways for getting the 5th card.
How do we find the number of items in neither of two sets?
Find the number of items in the union of the two sets and subtract from the number of items in the universe.
Which of the following does a graph search NOT include?
Finding the chromatic number
If a universal set is defined as all the countries of South America, which of the following is not an element of the universal set?
France
What kinds of graphs does Fleury's algorithm work for?
Graphs with an Euler path or circuit in it.
If he eats a hamburger, then he will eat two bags of fries. Which of the following represents the hypothesis in the above conditional statement?
He eats a hamburger.
Select the tautology from the statements below.
I will either pass my English class or not.
If the following two propositions are true, which is a true statement? Theresa loves dogs. Spot is a dog.
If Spot is a dog, then Theresa loves Spot.
Which of the following is a conditional statement?
If the dog barks, then the cat will meow.
In the adjacency matrix of an undirected graph, when is the value of a relationship equal to 1?
If there is an edge between two vertices
In the adjacency matrix of a directed graph, when is the value of a relationship between two vertices considered to be 0?
If there is no edge or, no directed edge between two vertices
What is the most specific term that describes the numbers 5, 10, 15, 20, 25, ...?
Infinite arithmetic sequence
Which of these is not true about Boolean logic?
It evaluates whether integers are TRUE or FALSE.
Finding the minimum spanning tree by adding links purely in order of weights, rejecting any links which produce a loop, is called _____.
Kruskal's Algorithm
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{3,3} ' an example of?
Loop
A two-dimensional rectangular array of numbers representing a shape's coordinates is properly called a:
Matrix
A matrix B multiplied by its identity matrix is equal to what?
Matrix B
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{3,4}' an example of?
Multi-edges
Which formula is equivalent to an OR gate?
NOR(NOR(X,Y),NOR(X,Y))
Which gate is equivalent to the circuit shown? Tuppo vako
NOT
Which of the methods is both optimal and efficient?
None
Simplify the Boolean expression as much as possible.
Not Q
Find a Hamilton circuit. 45123
Not possible.
Find the solution to the recurrence relation f(n) = 16 f(n/4) + n.
O(n2)
Find the solution to the recurrence relation f(n) = 4 f(n/8) + n2.
O(n2)
Find the solution to the recurrence relation f(n) = 8 f(n/2) + n3.
O(n3 logn)
Which of the following connectors is the logical disjunction?
OR
What is a logic gate?
One or more transistor switches used to convert a binary input into a binary output based on a logical operation
A given homogeneous linear recurrence has the following characteristic equation: s + 1 = 0. Which is the order of this homogeneous linear recurrence?
Order 1
What is the first step of mathematical induction?
Prove the first case, usually n = 1, is true.
Which of these is the first step in mathematical induction?
Prove the statement is true for the first element in the set.
Which of the following is a practical method to simplify Boolean functions of more than 6 variables?
Quine-McCluskey
Which of the following is not a tree search strategy?
Reorder
The turning of a figure or object about a fixed point is called a:
Rotation
For the directed graph shown, which relationship is INCORRECT?
S-->T
What is an example of a logical fallacy?
Saying something is true or false based on emotions
What do you need to do to prove the second step of mathematical induction?
Show that both sides of the statement equal each other.
Which of the following is the induction step in mathematical induction?
Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.
What is ' E1->{1,2} E2->{2,3} E3->{3,4} E4->{4,1} ' an example of?
Simple graph
What kinds of brackets are used in proper matrix notation?
Square brackets
What is the degree in which the variables are different from the mean called?
Standard deviation
What do you have to assume in mathematical induction?
The case n = k is true.
Knowing the generation function for the sequence of the natural numbers except 0 as below in the first row, how would you describe the sequence for the generation function in the second row?
The even natural numbers, except 0.
In the formula for calculating an arithmetic series, what does the following term represent? a1
The first term in the series
Which of the following makes a recursive function?
The function calls itself
When finding the number of items in either of two sets, why does it not work to simply add the number of items in each of the two individual sets?
The items that are in common with both sets will be counted too many times.
What is the chromatic number of a graph?
The minimum number of colors required to color a graph such that adjacent vertices have different colors.
Which statement is an example of circular reasoning?
The moon is made of cheese because the moon is made of cheese.
What does the solution to a recurrence for a divide-and-conquer algorithm estimate?
The number of operations needed to solve the problem
What are expected values?
The results we can anticipate given that a particular event occurs.
A universal set is defined as the set of natural numbers N. Therefore, U = N. Which of the following properly states in words subset A shown in the image.
The set of all natural numbers x such that x is less than 50.
What is the ratio of the desired outcome and the total number of possible outcomes?
Theoretical probability
Jeanette asks 20 people in her history class if they are comfortable using technology. Seven out of the 20 say they are comfortable using technology. Calculate the theoretical and the actual probability of this scenario.
Theoretical: 50% Actual: 35%
Assuming that an ace counts as a one, calculate the number of combinations of straight flushes for a poker hand. (In other words, disregard the royal flush.)
There are 36 possible combinations of straight flushes.
Calculate the number of combinations and the probability of getting a three-of-a-kind hand in poker.
There are 54,912 different combinations of three-of-a-kind hands in poker. There is approximately a 1 in 47 chance of getting a three-of-a-kind hand in poker.
Why are inverses important when it comes to isomorphisms?
They allow you to find the other value regardless of which value you are given
Which row would you multiply by -2 to help you find the inverse matrix? [100 012 001]
Third
Which of the following describes a loop?
This edge connects point C to point C.
The _____ represents the values by which the _____ must be moved.
Translation matrix, coordinate matrix
What is the negation of 'Triangles are not squares'?
Triangles are squares.
Which of the following Boolean expressions returns a false result?
True AND False
Transformations in math are:
Ways by which a shape, line, or point can be manipulated
When are vertices in a graph said to be adjacent?
When an edge passes between them.
Use De Morgan's Law to rewrite the negation of the statement: You get home by 10pm or you're grounded.
You don't get home by 10pm and you're not grounded.
What is the simplified expression of the following Boolean function: Z = ab'c'd' + abc'd' + a'b'cd' + ab'cd' + a'bcd' + abcd' + ab'c'd + abc'd + a'b'cd + ab'cd + a'bcd + abcd?
Z = a + c
Add the following matrices: [2 1 -2 4] + [-412 214]
[-213 018]
Subtract the following matrices: [1-2 4-6]-[1 -3 3 -9]
[0113]
What of these symbols is commonly used to enclose a matrix?
[]
Examples of a spanning tree would not include _____.
a shopping list
The sequence <1, 3, 5, 11, 21, 43, ... > has the recursive formula and characteristic equation stated below. Mark the answer that has its solution.
ai = 4/2
Adding an additional edge to a spanning tree would produce _____.
all of the above
Spanning trees are used in computer networks to assure _____.
all of the above
Find the rule for this series: 2 + 6 + 18 + 54 + ...
an = 2(3)n-1
Which equation below represents a geometric sequence?
an = 4(1/2)n-1
What do we call data that cannot be divided, which is distinct, and can only occur in certain values?
discrete data
The operation OR is also known as:
disjunction
A spanning tree connects nodes to the network _____.
exactly once
Finding the minimum spanning tree by listing all possible spanning trees, totaling their weights, and selecting the tree with the lowest total weight is called _____.
explicit enumeration
Which of the following functions is a surjection going from the set of real numbers to the set of real numbers?
f(x) = x2
Which of the following words means that a circle cannot be flipped over when determining the number of different possible arrangements of items?
fixed
Deleting unused data is called _____.
garbage collection
These two graphs are _____ because the second graph can be obtained from the first by dividing some edges of the first with more vertices.
homomorphic
Searching from left to right is a(n) _____ search strategy.
inorder
A game is played where you draw two cards from a deck of 52 cards. If the cards are both jacks, then you win $5. Otherwise you lose $1. If you play this game, then what can you expect to win or lose in the long run?
lose $0.97 per game
What variable represents the number of trials in an experiment?
n
Which variable represents the number of trials when solving the expected value formula?
n
Work to sort, organize the data, or maintain the Tree Sort is referred to as _____.
overhead
What variable represents the probability of success on an individual trial?
p
Which element is 'minimal' in the following diagram?
p
p: The pond is not frozen over. q: The fish are not jumping. For the combination p AND (NOT q), for which truth values of p and q is the combination true?
p: T q: F
p: The dog rolls over on command. q: The dog gets a treat. For the combination p AND q, for which truth values of p and q is the combination true?
p: T q: T
A search which starts at the tips (leaves) of the tree and works toward the root is the _____ strategy.
postorder
A search which starts at the root of the tree and bears left is the _____ strategy.
preorder
The decision tree structure begins at a node called the _____.
root
The formula for a finite geometric series is:
s = a1(1-rn)/1-r
The Fibonacci sequence, < 0, 1, 1, 2, 3, 5, 8, ...> has the recursive formula and initial values as stated below. Mark the correct characteristic equation for this version of the Fibonacci sequence.
s2-s-1 = 0
What is the formula for the standard deviation of a binomial random variable?
sqrt (n * P * (1-P))
Which of the following is an isomorphism?
tan 'x = sin x / cos x
If it is not possible to simplify a function using Karnaugh maps, this means that
the function might be simplified by another method or it is in its simplest form
What is the conclusion in the following statement? 'If p and q are odd integers, then pq is odd.'
then pq is odd.
Design a logic circuit means
transfering a human readable specification into a circuit diagram
Data sorted in a tree structure may occasionally need to be resorted to maintain _____.
tree symmetry
Which of the following is a linear recurrence relation?
un = 3 un - 1
An example of a recurrence relation is _____.
un = 3 un-1
If a value today is 6 times the value it was yesterday, the recurrence relation is _____.
un = 6 un-1
Which of the following is a first-order recurrence?
un = sin un - 1
What is NOT a step of the systematic procedure of designing a logic circuit
use different wire colors to connect the components
Discrete Math deals with problems concerning options which can be expressed as _____.
whole numbers
A spanning tree must connect nodes _____.
with a minimum number of edges
An agency decides to conduct a survey on household incomes in their county. Let x = the household income. What type of variable is x?
x is a continuous random variable.
You decide to collect a bunch of cans of soda and measure the volume of soda in each can. Let x = the number of mL of soda in each can. What type of variable is x?
x is a continuous random variable.
How do you traverse a graph in a breadth-first search?
By visiting all the adjacent vertices of a given vertex first, before moving to the next vertex.
What is the formula for binomial combinations?
C = n!/(n - x)!x!
suppose you are at a small get-together at a friend's house, and there are 13 people there, including yourself. When it comes to birthdays of the people at the party and the pigeonhole principle, which of the following statements must be true?
It must be true that at least two people share the same birthday month.
The best Hamilton circuit in a weighted graph is the one where the total cost is what?
Least
If someone asked you to choose between two candidates for president, but there were still five candidates running for president, what kind of logical fallacy is that?
Limited choice
At a particular casino, you pay $1 to play a hand of Blackjack. If you get a total of 21 in your first two cards, then you win $10. If not, you lose your dollar. If you are the only one playing this game, then how much do you expect to win or lose per hand?
Lose $0.52 per hand
Which one of the following pairs of sets is equivalent?
M = {red, green, brown, blue} N = {5, 6, 7, 8}
If a geometric series begins with the following term, what would the next term be?
a1 * r
What is the domain of √(x)?
0 to infinity
Which is the finite sequence of counting backwards from 3?
{3, 2, 1, 0}
Which is an infinite sequence of all our even numbers?
{2, 4, 6, 8, . . .}
Which is the finite sequence of the first four multiples of 9?
{9, 18, 27, 36}
U = {a, b, c, d, e, ..., z} Which of the following is not a subset of U?
{a, b, 1}
What is the correct way of denoting (or writing) the cardinality of set Q? Q = {orange, green, pink, red, black}
|Q| = 5
Which group of numbers does not appear to be a sequence with a set pattern?
4, -13, 1, 5, 16, ...
What does the statement, 'We will go get ice cream if and only if you clean your room' mean?
It means no clean room, then no ice cream.
Which of the following statements is FALSE?
A POSET is called a meet semilattice if every pair of elements has a 'least upper bound' element.
If a = c and b = c, what kinds of new statements can you make? A: If a = c and b = c, then a = b. B: If a = 1, then b = 1. C: If b = 2, then a = 1.
A and B
Which of the following is a logic proposition? A: Sam only eats square foods B: Once upon a time C: Circles roll
A and C
Which of the following connectors is the logical conjunction?
AND
Which is a correct logical disjunction formed with the following phrases? Algebra is easy. Geometry is a breeze.
Algebra is easy or geometry is a breeze.
Which of the following occurs with a direct proof?
All are correct
Which type of logical fallacy is illustrated below? The moon is made of cheese because it looks like cheese and no one has ever eaten the moon.
Appeal to ignorance
Which type of logical fallacy is illustrated below? Everyone in the class agrees that if x = 2 and y = 2, then x must equal y.
Appeal to popularity
Which type of logical fallacy is illustrated below? This chocolate pie must be bad because nobody is taking it.
Appeal to popularity
Which of the following is NOT necessary for a relation to be called a partially ordered relation?
Asymmetric relation
What kind of logic is presented below? Your dog loves rock and roll music because my dog loves rock and roll music.
Hasty generalization
Which of the following is the inverse of the below argument? If I am in Kansas, then I'm in the United States.
If I'm not in Kansas, then I'm not in the United States.
Which of the following is a conditional statement whose hypothesis is 'Joe has a red car' and whose conclusion is 'Billy gets to drive it'?
If Joe has a red car, then Billy gets to drive it.
Which is the correct definition of a disjunction?
It is when two statements are connected with an 'OR'. The combined compound statement can be labeled as true when just one of the statements is true.
Select the tautology from the statements below.
It will either rain or not rain.
If Jimmy does his chores, then Jimmy will get a big scoop of chocolate ice cream. Which of the following represents the conclusion in the above conditional statement?
Jimmy will get a big scoop of chocolate ice cream.
Which of the following statements is FALSE for the Cartesian product of two sets A and B?
Only one element of set A relates to an element of set B.
Let the universal set U be the set of Mr. Salada's 5th grade class of 17 boys and 13 girls. Let set A be the set of all the girls in Mr. Salada's class, and let set B be the set of all the boys in Mr. Salada's class. Which set has the most elements?
The complement of A
Which of these is always logically equivalent to the inverse?
The converse
The amount of money you spend on coffees every month can be calculated as a function of the number of drinks you order every month. What are the independent and dependent variables in this function?
The independent variable is the number of drinks you order per month and the dependent variable is the amount of money you spend on coffee per month.
A statement can be determined to be which of the following through a direct proof?
True or false
Let the universal set U be all the letters of the English alphabet. What is the complement of the empty set? (Note: the empty set is a subset of every set.)
U
In the following statement, which part is the hypothesis? If all mammals are animals, then dogs are animals.
all mammals are animals
What is the rule for the nth term of the geometric sequence if the third term is 96 and the fifth term is 1,536?
an = 6(4)n-1
If point A is found at (1,3) and is translated 6 units up, then the new coordinates would be:
(1,9)
According to Fleury's algorithm, how many odd vertices does a graph with an Euler circuit in it have?
0
How many vertices can be odd in a Eulerian graph?
0
Consider the following probability distribution (see table below) of the number of firearms in a household, constructed from a survey of 25,000 randomly selected households. Let X = the number of firearms in a household, and assume that the probability of a household having more than 6 firearms in the home is negligible. If a household is selected at random, then how many firearms would you expect them to have?
0.468
How many times are you allowed to pass an edge in an Euler path or circuit?
1
How many times do you visit a vertex when traveling either a Hamilton circuit or path?
1
If vertex 3 is your starting point, which vertex can you not go to?
1
If you started at vertex 1, at which vertex will you end up if you are traveling a Hamilton circuit?
1
Suppose you have a circular arrangement of three items. If the circle is free, in how many ways can the items be arranged?
1
The identity matrix can be likened to what number?
1
How is 3 represented as information in a computer?
1 1
The start and end points must be which two vertices? 1324
1 and 3
Which two vertices can you connect to Eulerize this graph? 45123
1 and 3
Vickie is playing a lottery where she will choose 3 numbers between 1 and 25. No numbers are repeated and the order of the numbers does not matter. What are her chances of picking the winning combination of numbers?
1 in 2,300
Suppose you draw two cards from a standard deck of 52. What is the probability that they are both kings?
1/221
5 C2
10
Alysha is at a local library with her best friend. They've noticed a lot of cute boys in the library, and Alysha bets that of the next twenty people to enter the library, exactly 12 will be cute boys. Assuming there is a 50% chance that the next person to walk into the library is a cute boy, what is the probability that Alysha will win the bet?
12%
A vehicle license plate uses three numbers and three letters on each plate. The numbers are listed first and then the letters. The numbers used range from 0-9 and the letters used can be any letter of the 26 letters of the alphabet. On any given license plate, the letters can be repeated, but the numbers cannot be repeated. How many different plates are possible?
12,654,720
A complete graph with 6 vertices has how many Hamilton circuits?
120
Jimmy is making multi-flavored ice cream cones by scooping in different flavors one at a time. Jimmy has 6 different flavors but can only put 3 flavors in each cone. The order of the flavors is important to him as it affects how he tastes each ice cream. How many different arrangements of cones can Jimmy make?
120
Evaluate h(x) = 5x + 1, when x=3.
16
The value at time n is the value at time (n - 1) plus 6. If the start value at time n = 0 is 4, the value at time n = 2 is _____.
16
Jane has cards with the letters ALABAMA on them. She lays the cards face down. What is the probability of Jane selecting two cards and both of them being a vowel?
2/7
Find the sum of the following geometric series:
2044
How many Euler paths are there in this graph? 1343
4
How many Euler paths does this graph have? 1342
4
Given the recurrence f(n) = 4 f(n/2) + 1, how many sub-problems will a divide-and-conquer algorithm divide the original problem into, and what will be the size of those sub-problems?
4 sub-problems, each of size n/2
Sam is working on a computer programming project. He is creating a game that allows the player to shoot a ball into a basket in competition with the computer. The computer is set to make 60% of the shots on the medium level. What is the probability that the computer will make exactly 3 baskets out of 10 shots?
4% probability of success
15 E n= 4 3n+ 5
402
8 different novels are to be placed side-by-side on a shelf. How many ways can the 8 novels be arranged on the shelf?
40320
Solve 8! (factorial)
40320
Consider an example with 100 students where 20 are taking discrete math, 30 are taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking discrete math and web design, 10 are taking Java and web design and 5 are taking all three classes. How many students are taking neither of the three classes?
44
A brick wall contains 52 bricks in its bottom row and 49 bricks in the next row up from the bottom row. Each subsequent row contains 3 fewer bricks than the row immediately below it. If the wall contains 16 rows, how many bricks total make up the wall?
472
Find the sum 1 + 8 + 15 + 22 + 29 using the formula for an arithmetic series. What number should you use to replace the variable n in the formula n/2(a1 +an)?
5
How many vertices are in this graph? figure
5
A class of 6 boys and 12 girls are preparing for finals. The teacher is going to select 3 students to be exempt from the test. What is the probability that the three students that are exempt will be all boys?
5/204
Suppose a survey was conducted across the country regarding the number of firearms that people had in their households. Let X = the number of firearms in a household. From the survey of 30,000 households it was determined that the empirical probability of X = 1 was 0.2. How many of the households in the survey had one firearm in their home?
6,000
Suppose a survey was conducted across the country regarding the number of firearms that people had in their households. If the sample size was 25,000 and 17,000 households reported that they had no firearms in their home, what would be the empirical probability that a randomly selected household had no firearms in their home?
68
What is the ratio of successful outcomes and the total number of trials?
Actual probability
Under what condition is the graph isomorphic?
All of the above answers are correct
Logic circuits can be used for:
All of these answers are correct.
Which of these questions should you ask yourself to help determine if mathematical induction is a good method to prove a given statement?
All of these are questions that could be asked when determining if mathematical induction is a good method of proof to use to prove a statement.
Which of the following statements about the pigeonhole principle is TRUE?
All of these statements about the pigeonhole principle are true.
The name one-to-one describes which function?
An injective function
If a graph has 0 odd vertices, at which vertex must you start?
Any vertex will work.
Which type of logical fallacy is illustrated in the example below? I will get an A in math class because I spent many sleepless nights studying.
Appeal to emotion
Using the table below, find the probability of 1 success out of 10 trials, with a probability of .5.
Approximately 1%
If the probability of a customer entering a store and purchasing a spoon tie is 35%, then what is the expected value of customers purchasing a spoon tie if 79 enter the store?
Approximately 28 customers
Using the table below, find the probability that there are 3 successes out of 10 trials, with a probability of .2.
Approximately 88%
What does ALU stand for?
Arithmetic logic unit
How is an adjacency list represented?
As a list of connected vertices
Suppose we wanted to use mathematical induction to prove that for each natural number n, 2 + 5 + 8 + ... + (3n - 1) = n(3n - 1) / 2. In our induction step, what would we assume to be true and what would we show to be true.
Assume: 2 + 5 + 8 + ... + (3k - 1) = k(3k - 1) / 2
If you're placing pigeons in pigeonholes, and there are more pigeons than pigeonholes, which of the following must be TRUE?
At least one pigeonhole will contain more than one pigeon.
What is the binomial probability formula?
B(x; n, P) = C * P^x * (1 - P)^(n - x)
Which method is efficient? A. Brute force B. Nearest neighbor C. Repeated nearest neighbor D. Cheapest link
B, C, and D
What is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure?
Binomial experiment
What is the number of successes in a binomial experiment called?
Binomial random variable
Which method is optimal?
Brute force
What is a database NOT required to do?
Correct Data.
When the probability is a combination of the possible successes that are less than or equal to the trial number, to what are we referring?
Cumulative binomial distribution
You conduct an experiment where you want to measure the number of rolls it takes to get two 6's in a row when you roll a fair six-sided die. State whether the random variable is discrete or continuous and give a summary of its values.
Discrete with values 2, 3, 4, 5, 6, etc.
Which of the following is NOT an aim of Boolean circuits simplification?
Dissipate more heat.
Which of the following is not a valid matrix operation?
Division
A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Which of the following represents the distance the ball bounces between the first and seventh bounces with sigma notation?
E = 6, n =1 2h(2/3)n
Which statement is false?
Every universal set has the same number of elements.
If an experiment has independent outcomes and a fixed number of trials, then what else will it need to be a binomial experiment?
Exactly two possible outcomes
What do we call the number of successful outcomes expected in an experiment?
Expected
Children born after the turn of the century have a 60% probability of needing braces. What are the expected value and standard deviation for a group of 30 children surveyed?
Expected Value: 18 Standard Deviation: 2.7
What is the number of successful outcomes expected in an experiment called?
Expected value
Which of these truth tables is correct for the XOR operation?
FTTF
What kind of reasoning uses two unrelated events to prove a point?
False cause
The Boolean NOT operator:
Flips the state (1 or 0) it applies to.
Which of the following is a conditional statement?
If a square is a rectangle, then a triangle is a shape.
Triangles have 180 degrees in total, and squares are two triangles put together. What can you say about the total degrees of a square?
If triangles have 180 degrees in total, then squares have 360 degrees in total.
If x = 1 and y = 2, what can be said about z if z = xy?
If z = xy, then z = 2.
What can be said about this graph? 1234
It has an Euler circuit.
What can be said about this graph?45123
It has an Euler path.
Which is the correct definition of a conjunction?
It is when two statements are connected with an 'AND'; only when both statements are true is the resulting compound statement true.
Suppose you play a game with two 4-sided dice with sides numbered 1 through 4. If you roll a sum of 8 (face down), you win $10. If you roll anything else, you lose $1. What can you expect to win or lose in this game?
Lose 31 cents per game
Justin is conducting an experiment. He wants to know which type of pet is most preferred among 2nd graders. He asks them if they prefer dogs, cats, or hamsters. Is this a binomial experiment?
No, there are more than two possible outcomes.
A binomial experiment must have two possible outcomes: success and failure. Select the answer that is an example of a binomial experiment.
Picking an ace (success) or not an ace (failure) out of a deck of cards. Replacing each of the cards before drawing again. Drawing 20 cards total.
Which of the following operations is NOT valid in Boolean math?
Subtraction
What are transistor switches?
Switches that manipulate binary numbers by opening and closing gates.
Consider the following inputs: c = 5; b = 8; d = 3. What does the following expression return? (c < b) OR (b < d)
TRUE
What is the difference between homogeneous linear recurrences and non-homogeneous linear recurrences?
The homogeneous linear recurrences express its elements exclusively as a function of its preceding elements, while the non-homogeneous linear recurrences may also express its elements as a function of the position i of the element.
A matrix multiplied by its inverse is equal to what?
The identity matrix.
Jeanette asks 50 students in her math class if they are comfortable using technology. Thirty-eight say they are comfortable using technology. Calculate the theoretical and actual probability of this scenario.
Theoretical: 50% Actual: 76%
Calculate the number of combinations of royal flushes.
There are 4 combinations of royal flushes, one for each suit.
How many different combinations of flushes are there in a fair deck of cards?
There are 5,108 combinations of flushes in a deck.
Calculate the number of combinations of getting a four-of-a-kind hand in poker.
There are 624 different combinations of four-of-a-kind hands in poker.
What is the term for the collection of elements from either of two sets?
Union
Ally is trying to determine if an experiment is binomial. She is asking twenty people in front of a pet store if dogs' mouths are cleaner than a toilet. She asks the participants to answer true or false to her question. As far as she knows, the responses of the previous participants do not influence the responses of the other participants. Could this be considered binomial?
Yes, there are two possible outcomes, the outcomes are independent, and there is a fixed number of trials.
Let F = {2, 5, 7, 9} Let G = {1, 4, 6, 8} Which of the following is true?
fUg= {1,2,4,5,6,7,8,9}
A spanning tree for n nodes uses _____ edges.
n - 1
The number of comparisons required to sort n items linearly varies as _____.
n-squared
The number of comparisons required to sort n items into a tree structure varies as _____.
nlog(n)
In a tree structure, the point at which a discrete decision is made is called a _____.
node
A minimum spanning tree is the tree with _____.
the minimum total weights
Which one of these sequences is a finite sequence?
{1, 2, 3, 4}
If a universal set is {1, 2, 3, 4, 5, 6, 7} and set C equals {1, 2, 3}, What is the complement of the complement of C?
{1, 2, 3}
Which is the infinite sequence where each number is the previous number times 3?
{1, 3, 9, 27, . . .}
How many propositions are there in the statement below? I will buy you a huge bacon cheeseburger and wash your car if and only if you give me your big screen television.
3
The converse of a logical statement is found by doing what?
Switching the hypothesis and the conclusion.